Continuity JEE Mathematics Notes

Core Identity and Overview

• Core identity of continuity at a point $x = a$:   $\lim_{x \to a} f(x) = f(a)$
• For a function to be continuous at $x = a$, three values must exist and be identically equal: the left-hand limit (LHL), the right-hand limit (RHL), and the actual function value $f(a)$.
• Key focus areas for problem-solving:
  • Piecewise functions and boundary point evaluations.
  • Determination of unknown parameters.
  • Modulus function $|x|$.
  • Greatest Integer Function (GIF) $[x]$.
  • Fractional part function $\{x\}$.
  • Signum function $\text{sgn}(x)$.
  • Application of standard limits.
  • Classification of discontinuities.
  • Intermediate Value Theorem (IVT) and Bolzano's Theorem.
  • Relationship between continuity and differentiability.

Limits Required for Continuity

• Meaning of a limit:
  • The expression $\lim_{x \to a} f(x) = L$ signifies that as $x$ approaches the value $a$, the corresponding output $f(x)$ approaches the value $L$.
  • The actual function value $f(a)$ does not need to exist for the limit $\lim_{x \to a} f(x)$ to exist.
  • Even when $f(a)$ exists, it is not required to equal the value of the limit for the limit itself to exist.
• Left-Hand Limit (LHL):
  • Represented as $\lim_{x \to a^-} f(x)$.
  • Describes the behavior of $f(x)$ as $x$ approaches $a$ through values strictly smaller than $a$.
  • Example: $x \to 2^-$ denotes approaching $2$ through values such as $1.9$, $1.99$, $1.999$, and so forth.
• Right-Hand Limit (RHL):
  • Represented as $\lim_{x \to a^+} f(x)$.
  • Describes the behavior of $f(x)$ as $x$ approaches $a$ through values strictly greater than $a$.
• Existence of a two-sided limit:
  • A finite two-sided limit $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits are finite and equal:     $\text{LHL} = \text{RHL}$
• Standard limits required for evaluation:
  • Trigonometric limit 1:     $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$
  • Trigonometric limit 2:     $\lim_{x \to 0} \frac{\tan(x)}{x} = 1$
  • Scaled trigonometric limit 1:     $\lim_{x \to 0} \frac{\sin(ax)}{x} = a$
  • Scaled trigonometric limit 2:     $\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}$
  • Cosine limit:     $\lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2}$
  • Exponential limit 1:     $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
  • Exponential limit 2:     $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$
  • Logarithmic limit:     $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1$
  • Power limit:     $\lim_{x \to 0} \frac{(1+x)^n - 1}{x} = n$
• Algebra of limits:
  • Limit laws for sum, difference, product, and quotient can be applied directly provided individual component limits exist.
  • For quotient operations, the limit of the denominator must be non-zero.

Definition and Conditions of Continuity

• Mathematical definition:
  • A function $f$ is defined as continuous at $x = a$ if and only if:     $\lim_{x \to a} f(x) = f(a)$
• The three necessary conditions for continuity at $x = a$:
  1. The function value $f(a)$ exists (i.e., $a$ belongs to the domain of $f$).
  2. The limit $\lim_{x \to a} f(x)$ exists, requiring $\text{LHL} = \text{RHL}$.
  3. The limiting value equals the actual function value: $\lim_{x \to a} f(x) = f(a)$.
• Master condition for continuity at $x = a$:   $\text{LHL} = \text{RHL} = f(a)$
• Distinction between limit and function value:
  • Consider the function:     $f(x) = \frac{x^2 - 1}{x - 1}$
  • For all $x \neq 1$, the algebraic expression simplifies to $x + 1$.
  • The limit as $x \to 1$ is:     $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$
  • However, the original function $f(x)$ is completely undefined at $x = 1$ because of division by zero.

Left and Right Continuity

• Left continuity definition:
  • A function $f$ is left-continuous at $x = a$ if:     $\lim_{x \to a^-} f(x) = f(a)$
• Right continuity definition:
  • A function $f$ is right-continuous at $x = a$ if:     $\lim_{x \to a^+} f(x) = f(a)$
• Full continuity requirement:
  • Full continuity requires simultaneous left and right continuity at $x = a$:     $\text{LHL} = f(a) = \text{RHL}$

Continuity on Intervals

• Continuity on an open interval $(a, b)$:
  • A function $f$ is continuous on $(a, b)$ if it is continuous at every individual point within the interval $(a, b)$.
• Continuity on a closed interval $[a, b]$:
  • Continuous at every interior point $x \in (a, b)$.
  • Right-continuous at the left endpoint $x = a$:     $\lim_{x \to a^+} f(x) = f(a)$
  • Left-continuous at the right endpoint $x = b$:     $\lim_{x \to b^-} f(x) = f(b)$
• Continuity on half-open intervals:
  • Requires ordinary continuity across all interior points.
  • Requires the appropriate one-sided continuity condition (left or right) at whichever endpoint is included in the set.

Discontinuity and Its Classification

• Removable discontinuity:
  • Definition: Occurs when the two-sided limit $\lim_{x \to a} f(x)$ exists (meaning $\text{LHL} = \text{RHL}$ is finite), but either $\lim_{x \to a} f(x) \neq f(a)$ or $f(a)$ is completely undefined.
  • Repair: This discontinuity can be removed or repaired simply by defining or redefining $f(a)$ to match the limiting value $\lim_{x \to a} f(x)$.
• Jump discontinuity:
  • Definition: Occurs when both $\text{LHL}$ and $\text{RHL}$ exist as finite values, but are not equal:     $\text{LHL} \neq \text{RHL}$
• Infinite discontinuity:
  • Definition: Occurs when $f(x)$ grows without bound (approaches $\infty$ or $-\infty$) as $x$ approaches $a$.
  • Example: The function $f(x) = \frac{1}{x - a}$ exhibits an infinite discontinuity at $x = a$, creating a vertical asymptote.
• Oscillatory discontinuity:
  • Definition: Occurs when the function oscillates endlessly between fixed bounds near a point without settling on a single limiting value.
  • Example: The function $f(x) = \sin\left(\frac{1}{x}\right)$ at $x = 0$.

Continuity of Standard Functions

• Polynomial functions:
  • Every polynomial function is continuous for all real numbers $x \in \mathbb{R}$.
• Rational functions:
  • A rational function $\frac{p(x)}{q(x)}$ (where $p$ and $q$ are polynomials) is continuous at all real numbers where the denominator $q(x) \neq 0$.
• Exponential functions:
  • Functions of the form $a^x$ (where $a > 0$) are continuous for all real numbers $x \in \mathbb{R}$.
• Logarithmic functions:
  • Functions of the form $\log_a(x)$ are continuous for all $x > 0$ in their domain.
• Trigonometric functions:
  • $\sin(x)$ and $\cos(x)$ are continuous for all real numbers $x \in \mathbb{R}$.
  • $\tan(x)$, $\cot(x)$, $\sec(x)$, and $\csc(x)$ are continuous at every point throughout their defined domains.
• Inverse trigonometric functions:
  • Every standard inverse trigonometric function (such as $\sin^{-1}(x)$, $\cos^{-1}(x)$, $\tan^{-1}(x)$) is continuous throughout its respective domain.
• Key habit for analysis:
  • Always verify the domain of a function before analyzing continuity; a function cannot be declared continuous at any point where it is undefined.

Modulus Function

• Piecewise definition:
  • $|x| = x$ for $x \ge 0$
  • $|x| = -x$ for $x < 0$
• Continuity properties:
  • The modulus function $|x|$ is continuous for all real numbers $x \in \mathbb{R}$.
  • Evaluation at $x = 0$:     $\text{LHL} = \lim_{x \to 0^-} (-x) = 0$$\text{RHL} = \lim_{x \to 0^+} (x) = 0$$f(0) = |0| = 0$$\text{LHL} = \text{RHL} = f(0) = 0$
• General composite modulus:
  • If a function $f(x)$ is continuous at a point, then its absolute value function $|f(x)|$ is also continuous at that point.
  • When solving equations involving $|f(x)|$, construct piecewise branches by splitting cases where $f(x) \ge 0$ and $f(x) < 0$
• Strategic focus:
  • For expressions containing $|f(x)|$, locate all real zeros of $f(x)$ to determine the boundary points for piecewise evaluation.

Greatest Integer Function

• Definition:
  • The Greatest Integer Function $[x]$ or $\text{floor}(x)$ evaluates to the greatest integer less than or equal to $x$.
  • Numerical examples:     $[2.7] = 2$$[3] = 3$$[-1.2] = -2$
• Behavior at integer points $n \in \mathbb{Z}$:
  • Left-hand limit:     $\lim_{x \to n^-} [x] = n - 1$
  • Right-hand limit:     $\lim_{x \to n^+} [x] = n$
• Continuity profile:
  • The function $[x]$ is continuous at every non-integer real number.
  • The function $[x]$ is discontinuous at every integer point $n \in \mathbb{Z}$.
• Composite GIF analysis:
  • For composite expressions [g(x)]\, identify all points where $g(x)$ approaches an integer value; these points are primary candidates for discontinuity.

Fractional Part Function

• Definition:
  • The fractional part function is defined as:     $\{x\} = x - [x]$
  • Range restriction:     $0 \le \{x\} < 1$
• Behavior at integer points $n \in \mathbb{Z}$:
  • Left-hand limit:     $\lim_{x \to n^-} \{x\} = 1$
  • Right-hand limit:     $\lim_{x \to n^+} \{x\} = 0$
  • Discontinuous at every integer point $n \in \mathbb{Z}$.
• Behavior between integers:
  • Behaves linearly between consecutive integers and is continuous throughout every open interval $(n, n+1)$.

Signum Function

• Piecewise definition:
  • $\text{sgn}(x) = -1$ for $x < 0$
  • $\text{sgn}(x) = 0$ for $x = 0$
  • $\text{sgn}(x) = 1$ for $x > 0$
• Continuity profile:
  • Evaluation at $x = 0$:     $\text{LHL} = \lim_{x \to 0^-} (-1) = -1$$\text{RHL} = \lim_{x \to 0^+} (1) = 1$
  • Since $\text{LHL} \neq \text{RHL}$, $\text{sgn}(x)$ has a jump discontinuity at $x = 0$.
  • The function is continuous at all non-zero real numbers $x \neq 0$

Algebra of Continuous Functions

• Let $f$ and $g$ be continuous functions at $x = a$:
  • Sum and Difference:     $f + g$ and $f - g$ are continuous at $x = a$
  • Product:     $f \times g$ is continuous at $x = a$
  • Constant Multiple:     $c \times f$ is continuous at $x = a$ for any real constant $c$
  • Quotient:     $\frac{f}{g}$ is continuous at $x = a$ provided $g(a) \neq 0$

Composite Functions

• Main Theorem:
  • If $g$ is continuous at $x = a$ and $f$ is continuous at $g(a)$, then the composite function $(f \circ g)(x) = f(g(x))$ is continuous at $x = a$
• Application examples:
  • $\sin(x^2 + 1)$: Inner function $g(x) = x^2 + 1$ is continuous everywhere; outer function $f(u) = \sin(u)$ is continuous everywhere. Thus, the composite is continuous everywhere.
  • $e^{\sin(x)}$: Continuous for all real $x$
  • $\log(1 + x^2)$: Inner function $1 + x^2 \ge 1 > 0$, continuous for all real $x$

Piecewise Functions

• Primary significance:
  • The boundary points (joining points) where the function changes algebraic definitions represent the primary candidate locations for discontinuity.
• Standard procedure for checking continuity at a joining point $x = a$:
  • Step 1: Identify the joining point $x = a$
  • Step 2: Calculate $\text{LHL}$ using the left-hand branch expression ($x < a$).
  • Step 3: Calculate $\text{RHL}$ using the right-hand branch expression ($x > a$).
  • Step 4: Calculate actual value $f(a)$ using the branch containing the equality ($x \ge a$ or $x \le a$).
  • Step 5: Impose the continuity equation:     $\text{LHL} = \text{RHL} = f(a)$
• Comprehensive Example:
  • Define piecewise function $f(x)$:     $f(x) = x^2 + k$ for $x < 1$$f(x) = 3x + 2$ for $x \ge 1$
  • Evaluate at joining point $x = 1$:     $\text{LHL} = \lim_{x \to 1^-} (x^2 + k) = 1 + k$$\text{RHL} = \lim_{x \to 1^+} (3x + 2) = 3(1) + 2 = 5$$f(1) = 3(1) + 2 = 5$
  • Impose continuity condition $1 + k = 5$:     $k = 4$
• Multiple joining points:
  • If a function transitions definitions at multiple distinct points (e.g., $x = a$ and $x = b$), carry out the five-step procedure independently for each point.

Finding Unknown Parameters

• Single parameter problems:
  • Set up the continuity relation $\text{LHL} = \text{RHL} = f(a)$ at the critical boundary point to create a single algebraic equation, then solve for the parameter.
• Multiple parameter problems:
  • Multiple joining points or extra constraints yield a system of simultaneous equations. Solve the linear or non-linear system for all unknown constants.
• Fast setup strategy:
  • Immediately write down the full statement $\text{LHL} = \text{RHL} = f(a)$ before carrying out algebraic manipulations.

Finding Points of Discontinuity

• Rational expressions:
  • Set the denominator expression equal to zero and identify all roots; check if those points lie in the function domain.
• Logarithmic expressions:
  • Determine where the argument is non-positive ($\text{argument} \le 0$) and evaluate boundary points.
• Square root expressions:
  • Apply domain constraint $\text{expression} \ge 0$ and evaluate points where the interior vanishes or fails definition.
• Greatest Integer Functions (GIF):
  • Locate all values of $x$ where the argument inside $[g(x)]$ evaluates to an integer.
• Piecewise definitions:
  • Test every individual point where the formula changes branches.
• Trigonometric denominators:
  • Locate points where denominator expressions like $\cos(x)$ or $\sin(x)$ become zero.

Intermediate Value Theorem

• Theorem Statement:
  • If $f$ is a continuous function on the closed interval $[a, b]$, and $N$ is any real number strictly between $f(a)$ and $f(b)$, then there exists at least one value $c \in [a, b]$ such that:     $f(c) = N$
• Geometric Interpretation:
  • A continuous function cannot transition from one value to another without taking on every single intermediate value between them.
• Root existence corollary:
  • If $f$ is continuous on $[a, b]$ and the endpoint values have opposite signs ($f(a)f(b) < 0$), then there exists at least one real root $c \in (a, b)$ such that $f(c) = 0$
• Key limitation:
  • The Intermediate Value Theorem proves the existence of at least one root; it does not calculate the numerical value of the root nor guarantee that the root is unique.

Bolzano's Theorem

• Theorem Statement:
  • If $f$ is continuous on the closed interval $[a, b]$ and $f(a)f(b) < 0$, then there exists at least one point $c \in (a, b)$ satisfying:     $f(c) = 0$
• Execution Procedure:
  1. Calculate endpoint values $f(a)$ and $f(b)$.
  2. Verify that $f$ is continuous on the entire interval $[a, b]$.
  3. Demonstrate a sign change across endpoints ($f(a)f(b) < 0$).
  4. Conclude that at least one real root exists in $(a, b)$.

Continuity Versus Differentiability

• Fundamental implication:
  • If a function $f$ is differentiable at $x = a$, then $f$ is guaranteed to be continuous at $x = a$:     $\text{Differentiable at } a \implies \text{Continuous at } a$
• Fallacy of the converse:
  • Continuity at a point does NOT imply differentiability at that point:     $\text{Continuous at } a \not\implies \text{Differentiable at } a$
• Standard counterexample:
  • The function $f(x) = |x|$ is continuous at $x = 0$ ($\text{LHL} = \text{RHL} = f(0) = 0$).
  • However, $f(x) = |x|$ is non-differentiable at $x = 0$ because the graph exhibits a sharp corner (the left derivative is $-1$ and the right derivative is $1$).

Graph-Based Questions

• Visual representation of discontinuity types on a graph:
  • Hole in graph: Represents a removable discontinuity.
  • Vertical jump: Indicates a jump discontinuity where left and right limits differ.
  • Vertical asymptote: Represents an infinite discontinuity where branches approach $\pm \infty$
  • Continuous oscillations: Represents an oscillatory discontinuity where the graph fails to settle on a single value.
• Information retrieval from graphs:
  • Determine $f(a)$ from the solid dot or filled point at $x = a$
  • Determine $\text{LHL}$ from the curve approaching $x = a$ from the left.
  • Determine $\text{RHL}$ from the curve approaching $x = a$ from the right.
  • Compare $\text{LHL}$, $\text{RHL}$, and $f(a)$ to establish continuity status.

Problem-Solving Algorithms

• Algorithm 1: Solving Parameter Problems for Continuity:
  1. Identify boundary / joining point $a$
  2. Evaluate $\text{LHL} = \lim_{x \to a^-} f(x)$
  3. Evaluate $\text{RHL} = \lim_{x \to a^+} f(x)$
  4. Evaluate actual value $f(a)$
  5. Equate $\text{LHL} = \text{RHL} = f(a)$
  6. Solve the algebraic equation(s) for the unknown parameter(s).
• Algorithm 2: Evaluating Discontinuities in Complex Expressions:
  1. Determine domain of the function.
  2. List candidate points for discontinuity (denominators vanishing, GIF arguments matching integers, branch transitions).
  3. Calculate one-sided limits only at suspect points.
• Algorithm 3: Handling GIF and Modulus Functions:
  1. For GIF [g(x)]\, locate integer crossings of $g(x)$.
  2. For modulus $|f(x)|$, locate real zeros of $f(x)$ for case-splitting.
• Algorithm 4: Root-Existence Problems:
  1. Verify continuity on interval $[a, b]$.
  2. Compute endpoint values $f(a)$ and $f(b)$.
  3. Check if $f(a)f(b) < 0$
  4. Apply IVT / Bolzano's Theorem to conclude root existence.

Master Formula Sheet

• Continuity condition at $x = a$:   $\lim_{x \to a} f(x) = f(a) \iff \text{LHL} = \text{RHL} = f(a)$
• Left continuity:   $\lim_{x \to a^-} f(x) = f(a)$
• Right continuity:   $\lim_{x \to a^+} f(x) = f(a)$
• Removable discontinuity condition:   $\text{LHL} = \text{RHL} \neq f(a) \quad \text{or} \quad f(a) \text{ is undefined}$
• Jump discontinuity condition:   $\text{LHL} \neq \text{RHL} \quad (\text{both limits finite})$
• Greatest Integer Function $[x]$:
  • Discontinuous at every integer $n \in \mathbb{Z}$
  • Continuous at every non-integer $x \notin \mathbb{Z}$
• Fractional Part Function $\{x\} = x - [x]$:
  • Discontinuous at every integer $n \in \mathbb{Z}$
• Differentiability-Continuity relation:   $\text{Differentiable} \implies \text{Continuous}$$\text{Continuous} \not\implies \text{Differentiable}$
• Bolzano's Theorem condition:   $\text{Continuous on } [a, b] \quad \text{and} \quad f(a)f(b) < 0 \implies \exists c \in (a, b) \text{ such that } f(c) = 0$

Final Checklist and Study Order

• Core Concepts Checklist:
  • Limits, LHL, RHL
  • Definition of continuity
  • One-sided continuity
  • Interval continuity
  • Classification of discontinuities
• Standard Functions Checklist:
  • Polynomials, rational, exponential, logarithmic, trigonometric, inverse trigonometric, modulus
• Special Functions Checklist:
  • GIF $[x]$, fractional part $\{x\}$, signum $\text{sgn}(x)$, composite forms
• Problem Types Checklist:
  • Piecewise functions, finding parameters, multiple joining points, graph-based questions, standard-limit continuity, domain-based discontinuities
• Theorems Checklist:
  • Algebra of continuous functions, composition theorem, IVT, Bolzano's theorem
• One-Minute Revision Summary:
  • Continuity at $a$: $\text{LHL} = \text{RHL} = f(a)$
  • Left continuity: $\text{LHL} = f(a)$
  • Right continuity: $\text{RHL} = f(a)$
  • Removable: $\text{LHL} = \text{RHL} \neq f(a)$ or $f(a)$ undefined
  • Jump: $\text{LHL} \neq \text{RHL}$
  • GIF: Discontinuous at all integers
  • Fractional part: $\{x\} = x - [x]$, discontinuous at all integers
  • Differentiability: $\text{Differentiable} \implies \text{Continuous}$
  • IVT: Continuous + intermediate value condition $\implies$ existence
  • Bolzano: Continuous + sign change ($f(a)f(b) < 0$) $\implies$ at least one root
• Recommended Study Sequence:
  1. Limits
  2. LHL / RHL
  3. Definition of Continuity
  4. Piecewise Functions
  5. Parameter Problems
  6. Modulus Function
  7. GIF / Fractional Part / Signum
  8. Standard Functions
  9. Composite Functions
  10. Types of Discontinuities
  11. IVT / Bolzano's Theorem
  12. Continuity + Differentiability