Continuity JEE Mathematics Notes

Vocabulary flashcards covering core conditions, types of discontinuities, special functions, standard limits, and key theorems from the Continuity JEE mathematics transcript.

Core Condition for Continuity

The master condition requiring that at $x = a$, $\text{LHL} = \text{RHL} = f(a)$, meaning the left-hand limit, right-hand limit, and actual function value exist and are equal.

Left-Hand Limit (LHL)

The limiting value $\lim_{x \to a^-} f(x)$ obtained as $x$ approaches $a$ through values strictly smaller than $a$.

Right-Hand Limit (RHL)

The limiting value $\lim_{x \to a^+} f(x)$ obtained as $x$ approaches $a$ through values strictly greater than $a$.

Left Continuity

The condition satisfied when a function meets $\lim_{x \to a^-} f(x) = f(a)$.

Right Continuity

The condition satisfied when a function meets $\lim_{x \to a^+} f(x) = f(a)$.

Removable Discontinuity

A discontinuity where the two-sided limit exists ($\text{LHL} = \text{RHL}$), but does not equal $f(a)$, or $f(a)$ is undefined.

Jump Discontinuity

A discontinuity where both $\text{LHL}$ and $\text{RHL}$ are finite, but $\text{LHL} \neq \text{RHL}$.

Infinite Discontinuity

A discontinuity occurring when $f(x)$ becomes unbounded near a point, typically resulting in a vertical asymptote.

Oscillatory Discontinuity

A discontinuity where the function oscillates endlessly near a point without settling on a single limiting value, such as $\sin\left(\frac{1}{x}\right)$ at $x = 0$.

Modulus Function

The function defined as $|x| = x$ for $x \ge 0$ and $|x| = -x$ for $x < 0$, which is continuous for all real $x$.

Greatest Integer Function (GIF)

The function $[x]$ or $\text{floor}(x)$ that returns the greatest integer less than or equal to $x$; it is continuous at non-integers and discontinuous at every integer.

Fractional Part Function

The function $\{x\} = x - [x]$ satisfying $0 \le \{x\} < 1$; it is continuous between integers and discontinuous at every integer.

Signum Function

The function $\text{sgn}(x)$ defined as $-1$ for $x < 0$, $0$ for $x = 0$, and $1$ for $x > 0$; it is discontinuous at $x = 0$ and continuous elsewhere.

Intermediate Value Theorem (IVT)

Theorem stating that if $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then there exists at least one $c \in [a,b]$ such that $f(c) = N$.

Bolzano's Theorem

Theorem stating that if $f$ is continuous on $[a,b]$ and $f(a)f(b) < 0$, then at least one root $c \in (a,b)$ satisfies $f(c) = 0$.

Differentiability vs Continuity Implication

The principle stating that differentiability at a point implies continuity at that point, but continuity does not imply differentiability.