List of top Mathematics Questions on Continuity and differentiability asked in JEE Main

If \[ f(x) = \begin{cases} \dfrac{a|x| + x^2 - 2(\sin|x|)(\cos|x|)}{x}, & x \ne 0, \\ b, & x = 0, \end{cases} \] is continuous at $ x = 0 $, then $ a + b $ is equal to.

Let \[ f(x)=\lim_{\theta\to 0}\frac{\cos(\pi x)-x^{2/\theta}\sin(x-1)}{1-x^{2/\theta}(x-1)}. \] Statement 1: $f(x)$ is discontinuous at $x=1$.
Statement 2: $f(x)$ is continuous at $x=-1$.

Let $\alpha,\beta\in\mathbb{R}$ be such that the function \[ f(x)= \begin{cases} 2\alpha(x^2-2)+2\beta x, & x<1 \\ (\alpha+3)x+(\alpha-\beta), & x\ge1 \end{cases} \] is differentiable at all $x\in\mathbb{R}$. Then $34(\alpha+\beta)$ is equal to}

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice-differentiable function such that $ f(2) = 1 $. If $ F(x) = x f(x) $ for all $ x \in \mathbb{R} $, and the integrals $ \int_0^2 x F'(x) \, dx = 6 $ and $ \int_0^2 x^2 F''(x) \, dx = 40 $, then $ F'(2) + \int_0^2 F(x) \, dx $ is equal to:

If $$ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6, \text{ then } f(1) \text{ is equal to:} $$

If the function $f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3},$ $x \in \mathbb{R} $, is continuous at $ x = 0 $, then $ f(0) $ is equal to:

Let $ f: (0, \pi) \to \mathbb{R} $ be a function given by
\[ f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\tan 8x / \tan 7x}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ \left(1 + |\cot x|\right)^{b^{\lfloor \tan x \rfloor}}, & \frac{\pi}{2} < x < \pi \end{cases} \]
Where $ a, b \in \mathbb{Z} $. If $ f $ is continuous at $ x = \frac{\pi}{2} $, then $ a^2 + b^2 $ is equal to __________.

Let $f: [-1, 2] \rightarrow \mathbb{R}$ be given by $f(x) = 2x^2 + x + [x^2] - [x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is:

If the function $ f(x) = \begin{cases} \frac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_e 2 \log_e 3, & x = 0 \end{cases} $ is continuous at $ x = 0 $, then the value of $ a^2 $ is equal to

For $a, b > 0$, let $ f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0, \\ \frac{x}{3}, & x = 0, \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a x \sqrt{x}}}, & x > 0 \end{cases} $ be a continuous function at $x = 0$. Then $\frac{b}{a}$ is equal to:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a function given by
\[f(x) = \begin{cases} \frac{1 - \cos 2x}{x^2}, & x < 0 \\\alpha, & x = 0, \text{ where } \alpha, \beta \in \mathbb{R}. \\\beta \sqrt{1 - \cos x} / x, & x > 0 \end{cases} \]
If $ f $ is continuous at $ x = 0 $, then $ \alpha^2 + \beta^2 $ is equal to:

Let $f: R -\{0,1\} \rightarrow R$be a function such that $f(x)+f\left(\frac{1}{1-x}\right)=1+x$ Then $f(2)$ is equal to

for some a,b,c ∈ $\N$, let f(x) = ax-3 and g(x)=xb+c, x ∈ $\R$. If (fog)^-1(x) = $(\frac{x-7}{2})^{\frac{1}{3}}$ then (fog) (ac) + (gof) (b) is equal to _________ .

If f(x) = [a+13 sinx] & x ε (0, $\pi$), then number of non-differentiable points of f(x) are [where 'a' is integer]

The number of points of non-differentiability of the function f(x) = [4 + 13sinx] in (0, 2𝜋) is ____.

Let f be a differentiable function such that x² f(x) - x = $4 \int^x_0tf(t)dt,f(1)=\frac{2}{3}. \;\text{Then 18f(3)is equal to}$

Let x = x(y) be the solution of the differential equation 2(y+2) log_e (y+2)dx+(x+4-2log_e(y+2))dy = 0, y >-1 with x (e⁴-2)=1. Then x(e⁹-2) is equal to

Let $x = 2$ be a root of the equation $x^2 + px + q = 0$ and $f(x)=\begin{cases}\frac{1-cos(x^2-4px+q^2+8q+16)}{(x-2p)^4} & x≠2p\\0 & x=2p\end{cases}$.
Then $lim_{x \rightarrow 2p}[f(x)]$, where [·] denotes greatest integer function, is

If the solution curve of the differential equation $ (y - 2 \log x) dx + (x \log x^2) dy = 0 $ passes through the points $ \left( e^{4/3}, \alpha \right) $ and $ \left( e^4, \alpha \right) $, then $ \alpha $ is equal to:

Let $ k $ and $ m $ be positive real numbers such that the function} \[ f(x) = \begin{cases} 3x^2 + \frac{k}{\sqrt{x} + 1}, & 0 < x < 1, \\ mx^2 + k^2, & x \geq 1 \end{cases} \] is differentiable for all $ x > 0 $. Then $ 8f'(8) \left(\frac{1}{f(8)}\right) $ is equal to __________

Let [x] denote the greatest integer function and f(x) = max{1+x+[x], 2+x, x+2[x]}, 0 ≤ x ≤2. Let m be the number of points in [0, 2], where f is not continuous and n be the number of points in (0, 2), where f is not differentiable. Then (m+n)² + 2 is equal to 2

The function $f : R→R$ defined by $f(x)=lim_{n→∞}\frac{cos2πx-x^{2n}sinx-1}{1+x^{2n+1}-x^{2n}}$ is continuous for all x in

If the minimum value of $f(x)=\frac{5 x^2}{2}+\frac{\alpha}{x^5}, x>0$, is 14 , then the value of $\alpha$ is equal to:

Let $f : R \rightarrow R$ be a continuous function such that $f (3x)- f(x)= x$. If $f(8)=7$, then $f (14)$ is equal to :

If $f(x)=\begin{cases} x+a, & x \leq 0 \\ |x-4|, & x\gt0\end{cases}$ and
$g(x)= \begin{cases}x+1 & , x\lt0 \\ (x-4)^2+b, & x \geq 0\end{cases}$
are continuous on $R$, then $(gof) (2)+(fog)(-2)$ is equal to: