Continuity and Differentiability

Introduction to Continuity and Differentiability

• Philosophical Context: Albert Einstein stated, "The whole of science is nothing more than a refinement of everyday thinking."

• Historical Figure: Sir Isaac Newton (1642-1727) is a central figure in the development of these mathematical concepts.

• Chapter Scope: This study guide covers the continuation of polynomial and trigonometric differentiation from Class XI, introducing:   - Continuity and Differentiability and the relations between them.   - Differentiation of inverse trigonometric functions.   - Exponential functions and logarithmic functions.   - Powerful differentiation techniques and fundamental theorems illustrated through differential calculus.

The Concept of Continuity

• Informal Understanding: Naively, a function is continuous at a fixed point if the graph of the function around that point can be drawn without lifting the pen from the paper.

• Introductory Instance of Discontinuity (Example 1): Consider $f(x)$ defined as:   - $f(x) = 1$ if $x \leq 0$   - $f(x) = 2$ if x > 0   - Analysis shows that at points to the left of $0$ (e.g., $-0.1, -0.01$), the value is $1$. At points to the right (e.g., $0.1, 0.01$), the value is $2$. The left hand limit ($1$) and right hand limit ($2$) do not coincide, making the function discontinuous at $x = 0$.

• Introductory Instance of Discontinuity (Example 2): Consider $f(x)$ defined as:   - $f(x) = 1$ if $x \neq 0$   - $f(x) = 2$ if $x = 0$   - Here, the left and right hand limits both equal $1$, but the value of the function at $x = 0$ is $2$. Since the limit does not coincide with the function value, it is discontinuous.

• Mathematical Definition (Definition 1): Suppose $f$ is a real function on a subset of the real numbers and let $c$ be a point in the domain of $f$. Then $f$ is continuous at $c$ if:   - $\lim_{x \to c} f(x) = f(c)$   - Elaborately: The left hand limit (LHL), right hand limit (RHL), and the function value at $x = c$ must exist and be equal to each other.

• Discontinuity: If $f$ is not continuous at $c$, it is said to be discontinuous at $c$, and $c$ is called a point of discontinuity of $f$.

Analytical Examples of Continuity

• Example 1: Polynomial at a Point: Check $f(x) = 2x + 3$ at $x = 1$.   - Value: $f(1) = 2(1) + 3 = 5$.   - Limit: $\lim_{x \to 1} (2x + 3) = 5$.   - Conclusion: Continuous at $x = 1$.

• Example 2: Square Function: Examine $f(x) = x^2$ at $x = 0$.   - Value: $f(0) = 0$.   - Limit: $\lim_{x \to 0} x^2 = 0$.   - Conclusion: Continuous at $x = 0$.

• Example 3: Modulus Function: Discuss $f(x) = |x|$ at $x = 0$.   - Defined as: $f(x) = -x$ if x < 0 and $f(x) = x$ if $x \geq 0$.   - LHL: $\lim_{x \to 0^-} (-x) = 0$.   - RHL: $\lim_{x \to 0^+} x = 0$.   - Value: $f(0) = 0$.   - Conclusion: Continuous at $x = 0$.

• Example 4: Pointwise Discontinuity: $f(x) = x^3 + 3$ if $x \neq 0$ and $f(x) = 1$ if $x = 0$.   - Limit as $x \to 0$ is $0^3 + 3 = 3$.   - Value is $1$.   - Conclusion: Not continuous because $3 \neq 1$.

• Example 5: Constant Function: $f(x) = k$ is continuous at every real number $c$ because $\lim_{x \to c} k = k = f(c)$.

• Example 6: Identity Function: $f(x) = x$ is continuous at every real number $c$ because $\lim_{x \to c} x = c = f(c)$.

Continuity in Domains and Intervals

• Definition 2: A real function $f$ is continuous if it is continuous at every point in its domain.

• Continuity on a Closed Interval [a, b]: For $f$ to be continuous on $[a, b]$, it must be continuous at every point in $(a, b)$, and:   - $\lim_{x \to a^+} f(x) = f(a)$   - $\lim_{x \to b^-} f(x) = f(b)$

• Singleton Domain: If the domain of $f$ is a single point, it is continuous there by definition.

• Polynomial Continuity (Example 14): Every polynomial function defined by $p(x) = a_0 + a_1 x + \dots + a_n x^n$ is continuous at every real number $c$ in its domain.

• Reciprocal Function (Example 9): $f(x) = \frac{1}{x}, x \neq 0$ is continuous at every point in its domain. To prove this, for any non-zero constant $c$:   - $\lim_{x \to c} \frac{1}{x} = \frac{1}{c} = f(c)$.

Analysis of Limits and Infinity

• Concept of Infinity: Analyzed via $f(x) = \frac{1}{x}$ near $x = 0$.

• Right Hand Limit (Table 5.1): As $x$ approaches $0$ from the right:   - Values: $1 \to 1, 0.1 \to 10, 0.01 \to 100, 10^{-n} \to 10^n$.   - Notation: $\lim_{x \to 0^+} f(x) = +\infty$.

• Left Hand Limit (Table 5.2): As $x$ approaches $0$ from the left:   - Values: $-1 \to -1, -0.1 \to -10, -0.01 \to -100, -10^{-n} \to -10^n$.   - Notation: $\lim_{x \to 0^-} f(x) = -\infty$.

• Key Distinction: Neither $+\infty$ nor $-\infty$ are real numbers; therefore, these limits do not exist in the context of real number continuity.

Greatest Integer Function Discontinuity

• Definition: $f(x) = [x]$ denotes the greatest integer less than or equal to $x$.

• Case 1: Non-integral points: For any real number $c$ that is not an integer, the limit as $x \to c$ equals $[c]$, matching the function value. It is continuous at these points.

• Case 2: Integral points: Let $c$ be an integer and r > 0 be small.   - $[c - r] = c - 1$   - $[c + r] = c$   - LHL: $\lim_{x \to c^-} f(x) = c - 1$.   - RHL: $\lim_{x \to c^+} f(x) = c$.   - Conclusion: Since $c - 1 \neq c$, the function is discontinuous at every integral point.

Algebra of Continuous Functions

• Theorem 1: Suppose $f$ and $g$ are real functions continuous at a real number $c$. Then:   1. $(f + g)$ is continuous at $x = c$.   2. $(f - g)$ is continuous at $x = c$.   3. $(f \cdot g)$ is continuous at $x = c$.   4. $(\frac{f}{g})$ is continuous at $x = c$, provided $g(c) \neq 0$.

• Rational Functions (Example 16): Since polynomial functions are continuous, rational functions defined by $f(x) = \frac{p(x)}{q(x)}$ (where $q(x) \neq 0$) are continuous at every point in their domain.

• Continuity of Sine (Example 17): Uses the fact $\lim_{x \to 0} \sin(x) = 0$.   - Proof: Let $x = c + h$. As $x \to c, h \to 0$.   - $\lim_{h \to 0} \sin(c + h) = \lim_{h \to 0} (\sin(c)\cos(h) + \cos(c)\sin(h)) = \sin(c)(1) + \cos(c)(0) = \sin(c)$.

• Continuity of Tangent (Example 18): $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Being a quotient of two continuous functions, it is continuous wherever $\cos(x) \neq 0$ (i.e., $x \neq (2n + 1)\frac{\pi}{2}$).

• Composite Function Continuity (Theorem 2): If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then the composite function $(f \circ g)$ is continuous at $c$.   - Example: $f(x) = \sin(x^2)$ is continuous as a composition of $g(x) = \sin(x)$ and $h(x) = x^2$.   - Example: $f(x) = |1 - x + |x||$ is continuous as a composition of absolute value and polynomial/modulus sums.

The Basics of Differentiability

• Definition: The derivative of $f$ at point $c$ is defined by:   - $f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}$

• Notation: Denoted as $f'(x)$, $\frac{d}{dx}(f(x))$, or $y'$ (if $y = f(x)$).

• Algebra of Derivatives Review:   1. $(u \pm v)' = u' \pm v'$   2. $(uv)' = u'v + uv'$ (Leibnitz or Product Rule)   3. $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$, where $v \neq 0$ (Quotient Rule)

• Standard Derivatives (Table 5.3):   - $\frac{d}{dx}(x^n) = nx^{n-1}$   - $\frac{d}{dx}(\sin(x)) = \cos(x)$   - $\frac{d}{dx}(\cos(x)) = -\sin(x)$   - $\frac{d}{dx}(\tan(x)) = \sec^2(x)$

• Requirement for Existence: A function is differentiable at $c$ if both the left hand derivative and right hand derivative are finite and equal.

Relation between Continuity and Differentiability

• Theorem 3: If a function $f$ is differentiable at a point $c$, then it is also continuous at that point.

• Proof:   - Since $f$ is differentiable at $c$, $\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = f'(c)$.   - For $x \neq c$, $f(x) - f(c) = \left(\frac{f(x) - f(c)}{x - c}\right)(x - c)$.   - $\lim_{x \to c} [f(x) - f(c)] = \lim_{x \to c} \left[\frac{f(x) - f(c)}{x - c}\right] \cdot \lim_{x \to c} [x - c] = f'(c) \cdot 0 = 0$.   - Hence, $\lim_{x \to c} f(x) = f(c)$.

• Important Caveat: The converse is not true. Every differentiable function is continuous, but not every continuous function is differentiable.

• Counter-example: f(x) = |x| at x = 0:   - Left hand limit of derivative: $\lim_{h \to 0^-} \frac{|0+h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1$.   - Right hand limit of derivative: $\lim_{h \to 0^+} \frac{h-0}{h} = 1$.   - Since $-1 \neq 1$, $|x|$ is not differentiable at $0$, despite being continuous.

The Chain Rule

• Definition (Theorem 4): If $f = v \circ u$, and $t = u(x)$, then:   - $\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}$

• Extended Chain Rule: For $f = (w \circ u) \circ v$, where $t = v(x)$ and $s = u(t)$, then:   - $\frac{df}{dx} = \frac{dw}{ds} \cdot \frac{ds}{dt} \cdot \frac{dt}{dx}$

• Implicit Functions Differentiation: When $x - y - \pi = 0$, we can re-evaluate as $y = x - \pi$. However, in forms like $y + \sin(y) = \cos(x)$, we differentiate directly with respect to $x$ using the chain rule on the variable $y$.   - Example: For $y + \sin(y) = \cos(x)$, differentiation yields:     - $\frac{dy}{dx} + \cos(y)\frac{dy}{dx} = -\sin(x)$     - $\frac{dy}{dx} = \frac{-\sin(x)}{1 + \cos(y)}$ where $y \neq (2n+1)\pi$.

Derivatives of Inverse Trigonometric Functions

• General Identity: Derivatives are found assuming they exist and applying the chain rule.

• Derivation for sin⁻¹ x:   - Let $y = \sin^{-1}(x)$, so $x = \sin(y)$.   - Differentiating w.r.t. $x$: $1 = \cos(y)\frac{dy}{dx}$.   - $\frac{dy}{dx} = \frac{1}{\cos(y)} = \frac{1}{\sqrt{1-\sin^2(y)}} = \frac{1}{\sqrt{1-x^2}}$.   - Defined for $x \in (-1, 1)$.

• Derivative Table:   - $\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$ for domain $(-1, 1)$.   - $\frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1-x^2}}$ for domain $(-1, 1)$.   - $\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$ for domain $\mathbb{R}$.

Exponential and Logarithmic Functions

• Growth Rate Discussion: Polynomial functions grow based on their degree ($x^{15}$ grows faster than $x^{10}$). However, exponential functions like $10^x$ grow faster than any polynomial function $x^n$.   - Evidence: If $x = 10^3$, $x^{100} = 10^{300}$, while $10^x = 10^{1000}$.

• Definition 3 (Exponential): The exponential function with base b > 1 is $y = b^x$.   - Features: Domain is $\mathbb{R}$, range is $\mathbb{R}^+$, passes through $(0, 1)$, ever increasing.   - Natural Exponential: Using the series sum $1 + \frac{1}{1!} + \frac{1}{2!} + \dots = e$ (approx $2.718$). The function is $y = e^x$.

• Definition 4 (Logarithm): $\log_b(a) = x$ if $b^x = a$.   - Common Logarithm: Base $10$.   - Natural Logarithm: Base $e$, denoted as $\ln$ or $\log$.   - Features: Domain is $\mathbb{R}^+$, range is $\mathbb{R}$, passes through $(1, 0)$, ever increasing.

• Logarithmic Properties:   1. Change of Base: $\log_a(p) = \frac{\log_b(p)}{\log_b(a)}$.   2. Product Rule: $\log_b(pq) = \log_b(p) + \log_b(q)$.   3. Power Rule: $\log_b(p^n) = n\log_b(p)$.   4. Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$.   5. Inversion: $x = e^{\log(x)}$ for x > 0.

• Theorems of Differentiation (Theorem 5):   - $\frac{d}{dx}(e^x) = e^x$   - $\frac{d}{dx}(\log(x)) = \frac{1}{x}$

Logarithmic Differentiation

• Purpose: Used for functions of the form $y = [u(x)]^{v(x)}$. Both functions must be positive.

• Process:   1. Take log on both sides: $\log(y) = v(x)\log(u(x))$.   2. Differentiate implicitly using chain and product rules: $\frac{1}{y}\frac{dy}{dx} = v(x)\frac{u'(x)}{u(x)} + v'(x)\log(u(x))$.   3. Multiply by $y$ to get the final derivative.

• Example (Example 28): Differentiate $a^x$.   - $\log(y) = x\log(a)$.   - $\frac{1}{y}\frac{dy}{dx} = \log(a)$.   - $\frac{dy}{dx} = a^x\log(a)$.

Parametric Forms and Second Order Derivatives

• Parametric Form: When $x = f(t)$ and $y = g(t)$, where $t$ is the parameter.   - $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$, provided $\frac{dx}{dt} \neq 0$.

• Example (Example 32): Find $\frac{dy}{dx}$ if $x = at^2, y = 2at$.   - $\frac{dx}{dt} = 2at$, $\frac{dy}{dt} = 2a$.   - $\frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t}$.

• Second Order Derivative: Denoted by $\frac{d^2y}{dx^2}$ or $f''(x)$ or $y_2$.   - Defined as $\frac{d}{dx}(\frac{dy}{dx})$.

• Higher Order Derivatives: Defined by repeating the differentiation process sequentially.

• Example (Example 35): Find $\frac{d^2y}{dx^2}$ if $y = x^3 + \tan(x)$.   - $\frac{dy}{dx} = 3x^2 + \sec^2(x)$.   - $\frac{d^2y}{dx^2} = 6x + 2\sec(x) \cdot \sec(x)\tan(x) = 6x + 2\sec^2(x)\tan(x)$.

Summary of Principles

• Continuity: A function is continuous if the limit equals the function value at a point. Sums, differences, products, and quotients of continuous functions remain continuous.

• Differentiability: Implies continuity, but not vice versa. Standard differentiation rules (Product, Quotient, Chain) apply.

• Inverse Trig: Derivatives involve algebraic expressions (e.g., $\frac{1}{1+x^2}$ for $\tan^{-1}(x)$).

• Exponentials/Logs: $e^x$ is invariant under differentiation; $\log(x)$ differentiates to its reciprocal $1/x$.

• Parametrics: Differentiation involves ratios of derivatives of the parameter.