Real Analysis

Types of discontinuity

A discontinuity occurs at a point where a function fails to be continuous, classified rigorously by the behavior of its left-hand and right-hand limits.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Classification of Discontinuities via Limits

Let $f: D \subset \mathbb{R} \to \mathbb{R}$ and let $c$ be a limit point of $D$. The function $f$ has a discontinuity at $c$ if $f$ is not continuous at $c$. Discontinuities are categorized into: (1) Removable Discontinuity if $\lim_{x \to c} f(x)$ exists finitely but $\lim_{x \to c} f(x) \neq f(c)$ (or $f(c)$ is undefined); (2) Jump Discontinuity (First Kind) if both left-hand limit $\lim_{x \to c^-} f(x)$ and right-hand limit $\lim_{x \to c^+} f(x)$ exist finitely but are unequal; (3) Essential Discontinuity (Second Kind) if at least one of the one-sided limits fails to exist finitely.

📐 Geometric Interpretation

Geometrically, a removable discontinuity appears as a missing point or a removable 'hole' in the graph. A jump discontinuity manifests as a sudden vertical gap or step where the curve breaks and continues at a different height. An essential discontinuity displays wild oscillations or vertical asymptotic behavior where the graph breaks down completely.

⚙️ Algebraic & Structural Meaning

Algebraically, discontinuity classification exposes the localized limit structure of functions. It demarcates domains where Riemann integration, differentiability, and sequence convergence theorems fail or must be restricted.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Existence of Left-Hand Limit	`Left neighborhood $(c-\delta, c)$`	Strictly Mandatory for First Kind Classification	If the left-hand limit does not exist finitely due to oscillation or divergence, the discontinuity cannot be classified as a simple jump and defaults to an essential discontinuity.
Existence of Right-Hand Limit	`Right neighborhood $(c, c+\delta)$`	Strictly Mandatory for First Kind Classification	If the right-hand limit fails to exist finitely, jump discontinuity criteria fail, pushing the classification into essential discontinuities.
Finiteness of Limit Values	`Neighborhood points`	Essential for removable vs essential distinction	If one-sided limits diverge to $\pm\infty$ (e.g., $f(x) = \sin(1/x)$ or $1/x$), the discontinuity is classified as essential rather than removable or jump.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Rational Functions & Removable Discontinuity

$$ f(x) = \frac{P(x)}{Q(x)}, \quad Q(c) = 0 \text{ and } P(c) = 0 $$

Rule: If both numerator and denominator vanish at $x = c$, cancel the common linear factor $(x-c)$. If the limit exists after cancellation, $x = c$ is a removable discontinuity.

Application: Directly tested in UP PGT to find the value to assign to $f(c)$ to make the function continuous.

Jump Magnitude Formula

$$ \text{Jump} = | \lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x) | $$

Rule: For step functions, floor functions, or fractional part functions, the jump at an integer point $c$ is precisely evaluated by taking the absolute difference of the one-sided limits.

Application: Frequently asked in DSSSB and KVS PGT vector/calculus integration limit problems.

Greatest Integer Function Discontinuities

$$ f(x) = [x] \text{ at } x = n \in \mathbb{Z} $$

Rule: The function $[x]$ has a jump discontinuity at every integer $n$, where $\lim_{x \to n^-} [x] = n-1$ and $\lim_{x \to n^+} [x] = n$, giving a jump magnitude of $1$.

Application: Standard recurring pattern across all State PGT mathematics papers.

Sine Oscillatory Discontinuity Test

$$ f(x) = \sin\left(\frac{1}{x}\right) \text{ at } x = 0 $$

Rule: When $x \to 0$, the argument diverges to infinity causing rapid bounded oscillations between $-1$ and $1$. Neither limit exists, confirming an essential discontinuity of the second kind.

Application: Standard conceptual trap in multiple-choice assertion questions.

Exponential Limit Discontinuity Check

$$ f(x) = \frac{1}{1 + a^{1/x}} $$

Rule: Evaluate left limit ($x \to 0^-$ gives $1/x \to -\infty$) and right limit ($x \to 0^+$ gives $1/x \to +\infty$) to instantly identify jump discontinuity values based on $a > 0$.

Application: Appears in advanced UP PGT calculus sorting problems.

Rapid Option Elimination via Limit Existence

$$ \text{If } \lim_{x \to c} f(x) \text{ does not exist, eliminate 'Removable'} $$

Rule: If the limit does not exist due to infinite divergence or oscillation, immediately cross out 'removable discontinuity' options to narrow down to jump or essential.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Removable with Jump Discontinuity

Common Pitfall: Candidates assume any function with a missing single point has a jump discontinuity.

Counterexample: $f(x) = \frac{x^2 - 4}{x - 2}$ for $x \neq 2$, and $f(2) = 5$

Why it fails: The limit as $x \to 2$ is $4$, which exists finitely. Since $\lim_{x \to 2} f(x) \neq f(2)$, it is a removable discontinuity, not a jump.

🛡️ Exam Defense: Always check if $\lim_{x \to c} f(x)$ exists finitely. If yes, and it doesn't match $f(c)$, it is removable.

⚠️ COMMON ERROR #02

Misidentifying Essential Discontinuity as Jump

Common Pitfall: Treating oscillating functions like $\sin(1/x)$ as having jump discontinuities.

Counterexample: $f(x) = \sin(1/x)$ at $x = 0$

Why it fails: Neither left nor right limit exists finitely because the function oscillates infinitely between $-1$ and $1$, failing jump criteria.

🛡️ Exam Defense: If a function oscillates infinitely near a point without settling on finite limits, classify it strictly as an essential discontinuity.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Removable Discontinuity	$\lim_{x \to c} f(x)$ exists finitely, but $\neq f(c)$	$L_1 = L_2 \neq f(c)$	A single missing point or isolated hole on the curve	Factor cancels out in numerator and denominator
Jump Discontinuity (First Kind)	Both $L_1$ and $L_2$ exist finitely but $L_1 \neq L_2$	$\|L_2 - L_1\| = \text{jump} > 0$	A vertical break/step between two distinct levels	Floor function $[x]$ or piecewise constant functions
Essential Discontinuity (Second Kind)	At least one one-sided limit fails to exist	Limit does not exist (infinity or oscillation)	Wild oscillation or vertical asymptote (blow-up)	trigonometric terms like $\sin(1/x)$ or $e^{1/x}$
Mixed Discontinuity	Combination of behaviors on subdomains	Piecewise limits differ	Multiple types across domain points	Piecewise defined functions with rational/irrational split

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Determine the type of discontinuity for the function $f(x) = \frac{\tan x}{x}$ at $x = 0$, given $f(0) = 2$.

(A) Removable discontinuity(B) Jump discontinuity(C) Essential discontinuity(D) Continuous at $x = 0$

📚 Method A: Formal Textbook Derivation

Step 1: Evaluate $\lim_{x \to 0} \frac{\tan x}{x} = 1$. Step 2: Note that $f(0) = 2$. Step 3: Since the limit exists finitely ($1$) but does not equal $f(0)$ ($2$), the discontinuity is removable.

⚡ Method B: 25-Second Exam Shortcut

Recognize standard limit $\lim_{x \to 0} \frac{\tan x}{x} = 1 \neq f(0)$. Instantly mark removable discontinuity.

💡 Tip: Check if redefining $f(0)$ to the limit value makes the function continuous.

CANONICAL PROBLEM 02 Dual-Method Breakdown

What type of discontinuity does $f(x) = e^{1/x}$ possess at $x = 0$?

(A) Removable discontinuity(B) Jump discontinuity(C) Essential discontinuity of second kind(D) Continuous

📚 Method A: Formal Textbook Derivation

Step 1: Compute $\lim_{x \to 0^-} e^{1/x} = 0$. Step 2: Compute $\lim_{x \to 0^+} e^{1/x} = +\infty$. Step 3: Since the right limit diverges to infinity, the limit does not exist finitely, confirming an essential discontinuity.

⚡ Method B: 25-Second Exam Shortcut

Exponential with reciprocal argument $1/x$ blowing up to infinity at $0$ directly indicates an essential discontinuity.

💡 Tip: Infinite one-sided limits automatically rule out removable and jump discontinuities.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The function $f(x) = \frac{x^2 - 1}{x - 1}$ for $x \neq 1$ and $f(1) = 3$ has at $x = 1$: \n (A) Removable discontinuity \n (B) Jump discontinuity \n (C) Essential discontinuity \n (D) Continuity

A Removable discontinuity

B Jump discontinuity

C Essential discontinuity

D Continuity

Derivation & Analysis:

Limit as $x \to 1$ is $\lim_{x \to 1} (x+1) = 2$. Since $f(1) = 3 \neq 2$, the discontinuity is removable.

MCQ 2: The function $f(x) = \frac{1}{1 + 2^{1/x}}$ at $x = 0$ exhibits: \n (A) Removable discontinuity \n (B) Jump discontinuity \n (C) Essential discontinuity \n (D) Continuous

A Removable discontinuity

B Jump discontinuity

C Essential discontinuity

D Continuous

Derivation & Analysis:

As $x \to 0^-$, $1/x \to -\infty$, $2^{1/x} \to 0$, so limit is $1/(1+0) = 1$. As $x \to 0^+$, $1/x \to +\infty$, $2^{1/x} \to \infty$, so limit is $0$. Unequal finite limits imply a jump discontinuity.

MCQ 3: The greatest integer function $f(x) = [x]$ at $x = 3$ has: \n (A) Removable discontinuity \n (B) Jump discontinuity of magnitude $1$ \n (C) Essential discontinuity \n (D) Continuity

A Removable discontinuity

B Jump discontinuity of magnitude $1$

C Essential discontinuity

D Continuity

Derivation & Analysis:

Left limit is $2$ and right limit is $3$. The jump is $3 - 2 = 1$, representing a jump discontinuity.

MCQ 4: The function $f(x) = \sin\left(\frac{1}{x}\right)$ at $x = 0$ has: \n (A) Removable discontinuity \n (B) Jump discontinuity \n (C) Essential discontinuity \n (D) No discontinuity

A Removable discontinuity

B Jump discontinuity

C Essential discontinuity

D No discontinuity

Derivation & Analysis:

Since neither one-sided limit exists due to infinite oscillation between $-1$ and $1$, it is an essential discontinuity.

MCQ 5: If $\lim_{x \to c^-} f(x) = L_1$ and $\lim_{x \to c^+} f(x) = L_2$ with $L_1 \neq L_2$ both being finite, the discontinuity is: \n (A) Removable \n (B) Jump (First Kind) \n (C) Essential (Second Kind) \n (D) None

A Removable

B Jump (First Kind)

C Essential (Second Kind)

D None

Derivation & Analysis:

By definition, unequal finite one-sided limits constitute a jump discontinuity of the first kind.

MCQ 6: The function $f(x) = \frac{\sin x}{x}$ for $x \neq 0$ and $f(0) = 1$ is: \n (A) Discontinuous (Removable) \n (B) Continuous everywhere \n (C) Discontinuous (Jump) \n (D) Discontinuous (Essential)

A Discontinuous (Removable)

B Continuous everywhere

C Discontinuous (Jump)

D Discontinuous (Essential)

Derivation & Analysis:

Since $\lim_{x \to 0} \frac{\sin x}{x} = 1 = f(0)$, the function is continuous.

MCQ 7: Which of the following functions has an essential discontinuity at $x = 0$? \n (A) $f(x) = x \sin(1/x)$ \n (B) $f(x) = \cos(1/x)$ \n (C) $f(x) = \frac{|x|}{x}$ \n (D) $f(x) = \frac{x^2}{x}$

A $f(x) = x \sin(1/x)$

B $f(x) = \cos(1/x)$

C $f(x) = \frac{|x|}{x}$

D $f(x) = \frac{x^2}{x}$

Derivation & Analysis:

$\cos(1/x)$ oscillates infinitely between $-1$ and $1$ as $x \to 0$ without limit, making it an essential discontinuity. Note: $x \sin(1/x)$ has a removable discontinuity at $0$ if defined.

MCQ 8: Assertion (A): $f(x) = \frac{1}{x-2}$ has an essential discontinuity at $x = 2$. Reason (R): The one-sided limits as $x \to 2$ are infinite. \n (A) Both A and R are true and R is correct explanation \n (B) Both true but R is not explanation \n (C) A is true, R is false \n (D) A is false, R is true

A Both A and R are true and R is correct explanation

B Both true but R is not explanation

C A is true, R is false

D A is false, R is true

Derivation & Analysis:

Infinite one-sided limits mean the limit does not exist finitely, classifying it as an essential discontinuity of the second kind.

MCQ 9: The fractional part function $f(x) = \{x\}$ has discontinuities at: \n (A) All integers \n (B) No points \n (C) All rational numbers \n (D) All irrational numbers

A All integers

B No points

C All rational numbers

D All irrational numbers

Derivation & Analysis:

The fractional part function $\{x\} = x - [x]$ jumps at every integer, producing jump discontinuities at all $x \in \mathbb{Z}$.

Q10

MCQ 10: If a function $f$ has a removable discontinuity at $x = c$, then: \n (A) $\lim_{x \to c} f(x)$ does not exist \n (B) $\lim_{x \to c} f(x) = f(c)$ \n (C) $\lim_{x \to c} f(x)$ exists finitely but $\neq f(c)$ \n (D) One-sided limits are infinite

A $\lim_{x \to c} f(x)$ does not exist

B $\lim_{x \to c} f(x) = f(c)$

C $\lim_{x \to c} f(x)$ exists finitely but $\neq f(c)$

D One-sided limits are infinite

Derivation & Analysis:

By definition, a removable discontinuity requires the limit to exist finitely while differing from the function value at that point.

Common questions

Yes. By defining $f(c)$ to equal the finite limit $\lim_{x \to c} f(x)$, the discontinuity is successfully removed, making the function continuous at $x = c$.

Historical terminology classifies discontinuities where both left and right-hand limits exist finitely as 'first kind', whereas those where at least one limit fails to exist are 'second kind' (essential).

Immediately evaluate $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$. If both are finite and equal (but $\neq f(c)$), it is removable. If both are finite and unequal, it is jump. If either limit is infinite or oscillates, it is essential.

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