Calculus

Continuity & Differentiability

Easy

common

If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$ , then which of the following can be a discontinuous function ?

$f(x) + g(x)$

$f(x) - g(x)$

$f(x) \cdot g(x)$

$\frac{g(x)}{f(x)}$

✅ Correct Option: 4

Both $f(x)=2x$ and $g(x)=x^2/2+1$ are polynomials, hence continuous. Sum, difference, and product of continuous functions remain continuous.

The quotient $g(x)/f(x)=(x^2/2+1)/(2x)$ is undefined at $x=0$ , where $f(x)=0$ , making it discontinuous there.

2026: 12 May Shift 2

If $y = \dfrac{\log_e x}{x}$ , then $\dfrac{d^2y}{dx^2}$ is :

2025: 22 May Shift 2

Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

List-I	List-II
(A) $\sin x$ is continuous on:	(I) $\mathbb{R} - \{0\}$
(B) $\tan x$ is continuous on:	(II) $\mathbb{R}$
(C) $\cot x$ is continuous on:	(III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$
(D) $x^{-n}, n \in \mathbb{N}$ is continuous on:	(IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

Choose the correct answer from the options given below:

2026: 14 May Shift 2

If $f(x)=\begin{cases}\frac{\sin(a-b)x}{x}, & x<0\\ 2a+b, & x=0\\ \frac{-\tan 12x}{bx}, & x>0\end{cases}$ , $(b>0)$ is continuous at $x=0$ , then

2025: 29 May Shift 2

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

2025: 3 June Shift 2

If $x^m y^n = (x + y)^{m+n}$ , then $\frac{d^2y}{dx^2}$ is equal to:

2024: 16 May Shift 1

Choose the correct answer from the options given below :

If f(x)=2xf(x) = 2xf(x)=2x and g(x)=x22+1g(x) = \frac{x^2}{2} + 1g(x)=2x2​+1, then which of the following can be a discontinuous function ?