Continuity and Differentiability — JEE Main PYQs

143 previous year questions on Continuity and Differentiability (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let $[∙]$ denote the greatest integer function, and let $f (x) = min {2 x, x^{2}}$ . Let $S = {x \in (- 2, 2) :$ the function $g (x) = ∣ x ∣ [x^{2}]$ is discontinuous at x $}$ .

Then $\sum_{x \in S} f (x)$ equa …
JEE Main 2026View solution →
Let $f (x) = {\frac{a x ^{2} + 2 a x + 3}{4 x ^{2} + 4 x - 3}, b x \neq = - \frac{3}{2}, \frac{1}{2} x = - \frac{3}{2}, \frac{1}{2}$
be continuous at $x = - \frac{3}{2}$ . If $fof (x) = \frac{7}{5}$ , then $x$ is equal to :
JEE Main 2026View solution →
If $f (x) = {\frac{a ∣ x ∣ + x ^{2} - 2 ( s i n ∣ x ∣ ) ( c o s ∣ x ∣ )}{x} b, x \neq = 0, x = 0$
is continuous at $x = 0$ , then $a + b$ is equal to :
JEE Main 2026View solution →
If the function $f (x) = \frac{e ^{x} ( e ^{t a n x - x} - 1 ) + l o g _{e} ( s e c x + t a n x ) - x}{t a n x - x}$ is Continuous at $x = 0$ , then the value of $f (0)$ is equal to
JEE Main 2026View solution →
Let $α, β \in R$ be such that the function
$f (x) = {2 α (x^{2} - 2) + 2 β x (α + 3) x + (α - β), x < 1, x \geq 1$
be differentiable at all …
JEE Main 2026View solution →
Consider the following three statements for the function $f : (0, \infty) \to R$ defined by
$f (x) = ∣ lo g_{e} x ∣ - ∣ x - 1∣ :$
(I) f is differentiable at all $x > 0$ .
(II) f is increasing in …
JEE Main 2026View solution →
Let $y = y (x)$ be a differentiable function in the interval $(0, \infty)$ such that $y (1) = 2$ .
and …
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Let $[t]$ denote the greatest integer less than or equal to $t$ . If the function …
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Let $f (x) = lim_{θ \to 0} (\frac{c o s π x - x ^{(\frac{2}{θ})} s i n ( x - 1 )}{1 + x ^{(\frac{2}{θ})} ( x - 1 )}), x \in R$ .

Consider the following two statements :
(I) $f (x)$ is discontinous at $x = 1$ .
(I …
JEE Main 2026View solution →
Let $f (x) = {e^{x - 1} x^{2} - 5 x + 6, x < 0, x \geq 0$ and
$g (x) = f (∣ x ∣) + ∣ f (x) ∣$ . If the number of points where g is not continuous and is not differ …
JEE Main 2026View solution →
Let f be a real polynomial of degree n such that $f (x) = f^{'} (x) f^{''} (x)$ , for all $x \in R$ . If $f (0) = 0$ , then $36 (f^{'} (2) + f^{''} (2) + \int_{0}^{2} f (x) d x)$ is equal to :
JEE Main 2026View solution →
The number of points, at which the fucntion
$f (x) = max {6 x, 2 + 3 x^{2}} + ∣ x - 1∣ cos x^{2} - \frac{1}{4}, x \in (- π, π)$

is not differentiable, is ______.
JEE Main 2026View solution →
Let $f (x)$ and $g (x)$ be twice differentiable functions satisfying $f^{''} (x) = g^{''} (x)$ for all $x \in R$ , $f^{'} (1) = 2 g^{'} (1) = 4$ and …
JEE Main 2026View solution →
Let $f (x) = {x^{3} + 8 x^{2} - 4 x < 0 x \geq 0$ and
$g (x) = {(x - 8)^{1/3}; (x + 4)^{1/2}; x < 0 x \geq 0$
Then the number of points, where the function …
JEE Main 2026View solution →
For the function $f (x) = e^{s i n ∣ x ∣} - ∣ x ∣, x \in R$ , consider the following statements:

Statement I : f is differentiable for all $x \in R$ .
Statement II : f is increasing in …
JEE Main 2026View solution →
Let $f (x) = {\frac{1}{3} \frac{b ( 1 - s i n x )}{( π - 2 x ) ^{2}},, x \leq π /2 x > π /2$ , If $f$ is continuous at $x = π /2$ , then the value of …
JEE Main 2026View solution →
The number of points in the interval [2,4], at which the function $f (x) = [x^{2} - x - \frac{1}{2}]$ , where $[\cdot]$ denotes the greatest integer function, is discontinuous, is ____ .
JEE Main 2026View solution →
If the function
$f (x) = ⎩ ⎨ ⎧ \frac{2}{x} {sin (k_{1} + 1) x + sin (k_{2} - 1) x} 4 \frac{2}{x} lo g_{e} (\frac{2 + k _{1} x}{2 + k _{2} x}), x < 0, x = 0, x > 0$
is continuous at $x = 0$ , then …
JEE Main 2025View solution →
Let
$f (x) = ⎩ ⎨ ⎧ 3 x, min {1 + x + [x], x + 2 [x]}, 5, x < 0 0 \leq x \leq 2 x > 2$
where [.] denotes greatest integer function. If $α$ and $β$ are the number o …
JEE Main 2025View solution →
Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points, where the function $f (x) = [x] + ∣ x - 2∣$ , $- 2 < x < 3$ , is not continuous and not differentiable.

Then …
JEE Main 2025View solution →
Let $f : R \to R$ be a twice differentiable function such that $(sin x cos y) (f (2 x + 2 y) - f (2 x - 2 y)) = (cos x sin y) (f (2 x + 2 y) + f (2 x - 2 y))$ , for all $x, y \in R$ .
If $f^{'} (0) = \frac{1}{2}$ , then the value of $24 f^{''} (\frac{5 π}{3})$ is:
JEE Main 2025View solution →
Let $f (x) = ⎩ ⎨ ⎧ (1 + ax)^{1/ x} 1 + b \frac{( x + 4 ) ^{1/2} - 2}{( x + c ) ^{1/3} - 2},,, x < 0 x = 0 x > 0$
be continuous at $x = 0$ . Then …
JEE Main 2025View solution →
Let $m$ and $n$ be the number of points at which the function $f (x) = max {x, x^{3}, x^{5}, \dots, x^{21}}$ , $x \in R$ , is not differentiable and not continuous, respectively. Then $m + n$ is equal to ______.
JEE Main 2025View solution →
If the function $f (x) = \frac{t a n ( t a n x ) - s i n ( s i n x )}{t a n x - s i n x}$ is continuous at $x = 0$ , then $f (0)$ is equal to ______.
JEE Main 2025View solution →
The number of points of discontinuity of the function $f (x) = [\frac{x ^{2}}{2}] - [x]$ , $x \in [0, 4]$ , where $[\cdot]$ denotes the greatest integer function is
JEE Main 2025View solution →
Let $f : R \to R$ be a function given by $f (x) = ⎩ ⎨ ⎧ \frac{1 - c o s 2 x}{x ^{2}}, α, \frac{β 1 - c o s x}{x}, x < 0 x = 0 x > 0$ where $α, β \in R$ . If f is continuous at $x = 0$ , then $α^{2} + β^{2}$ is equal to
JEE Main 2024View solution →
If the function $f (x) = ⎩ ⎨ ⎧ \frac{7 2 ^{x} - 9 ^{x} - 8 ^{x} + 1}{2 - 1 + cos x}, a lo g_{e} 2 lo g_{e} 3, x \neq = 0 x = 0$ is continuous at $x = 0$ , then the value of $a^{2}$ is equal to
JEE Main 2024View solution →
If the function $f (x) = \frac{s i n 3 x + α s i n x - β c o s 3 x}{x ^{3}}, x \in R$ , is continuous at $x = 0$ , then $f (0)$ is equal to:
JEE Main 2024View solution →
Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f (1) = 1$ and $lim_{t \to x} \frac{t ^{2} f ( x ) - x ^{2} f ( t )}{t - x} = 1$ for each $x > 0$ . Then $2 f (2) + 3 f (3)$ is equal to ____.
JEE Main 2024View solution →
Let $f : [- 1, 2] \to R$ be given by $f (x) = 2 x^{2} + x + [x^{2}] - [x]$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . The number of points, where $f$ is not continuous, is
JEE Main 2024View solution →
If $f (x) = {x^{3} sin (\frac{1}{x}), 0, x \neq = 0 x = 0$ , then
JEE Main 2024View solution →
Let $[t]$ denote the greatest integer less than or equal to $t$ . Let $f : [0, \infty) \to R$ be a function defined by $f (x) = [\frac{x}{2} + 3] - [x]$ . Let $S$ be the set of all points in the interval $[0, 8]$ at which $f$ is not continuous. Then …
JEE Main 2024View solution →
For $a, b > 0$ , let …
JEE Main 2024View solution →
Let $f : (0, π) \to R$ be a function given by
$f (x) = ⎩ ⎨ ⎧ (\frac{8}{7})^{\frac{t a n 8 x}{t a n 7 x}}, a - 8, (1 + ∣ cot x ∣)^{\frac{b}{a} ∣ t a n x ∣}, 0 < x < \frac{π}{2} x = \frac{π}{2} \frac{π}{2} < x < π$
…
JEE Main 2024View solution →
Consider the function $f : (0, \infty) \to R$ defined by $f (x) = e^{- ∣ l o g_{e} x ∣}$ . If m and n be respectively the number of points at which f is not continuous and f is not differentiable, then …
JEE Main 2024View solution →
If the function $f (x) = {\frac{1}{∣ x ∣} a x^{2} + 2 b,, ∣ x ∣ \geq 2 ∣ x ∣ < 2$ is differentiable on $R$ , then $48 (a + b)$ is equal to ______.
JEE Main 2024View solution →
Consider the function $f : (0, 2) \to R$ defined by $f (x) = \frac{x}{2} + \frac{2}{x}$ and the function $g (x)$ defined by $g (x) = {min {f (t)}, \frac{3}{2} + x, 0 < t \leq x and 0 < x \leq 1 1 < x < 2$ . Then
JEE Main 2024View solution →
Consider the function.
$f (x) = ⎩ ⎨ ⎧ \frac{a ( 7 x - 12 - x ^{2} )}{b ∣ x ^{2} - 7 x + 12 ∣} 2^{\frac{s i n ( x - 3 )}{x - [ x ]}} b, x < 3, x > 3, x = 3$

Where $[x]$ denotes the greatest integer less than or equal to $x$ . …
JEE Main 2024View solution →
Let $f (x) = 2 x^{2} + 5 x ∣ - 3∣, x \in R$ . If m and n denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to :
JEE Main 2024View solution →
Let $g (x)$ be a linear function and $f (x) = {g (x) (\frac{1 + x}{2 + x})^{\frac{1}{x}}, x \leq 0, x > 0$ , is continuous at $x = 0$ . If $f^{'} (1) = f (- 1)$ , then the value of $g (3)$ is
JEE Main 2024View solution →
Let $f : R \to R$ be defined as
$f (x) = ⎩ ⎨ ⎧ \frac{a - b c o s 2 x}{x ^{2}} x^{2} + c x + 2 2 x + 1;;; x < 0 0 \leq x \leq 1 x > 1$
If $f$ is continuous everywhere in $R$ and m is the number of points where $f$ …
JEE Main 2024View solution →
Let a and b be real constants such that the function $f$ defined by $f (x) = {x^{2} + 3 x + a, b x + 2, x \leq 1 x > 1$ be differentiable on $R$ . Then, the value of $\int_{- 2}^{2} f (x) d x$ equals
JEE Main 2024View solution →
Let $f : R - {0} \to R$ be a function satisfying
$f (\frac{x}{y}) = \frac{f ( x )}{f ( y )}$ for all $x, y, f (y) \neq = 0$ . If …
JEE Main 2024View solution →
Let $a \in Z$ and $[t]$ be the greatest integer $\leq t$ . Then the number of points, where the function $f (x) = [a + 13 sin x], x \in (0, π)$ is not differentiable, is ____
JEE Main 2023View solution →
Let k and m be positive real numbers such that the function $f (x) = {3 x^{2} + k x + 1, m x^{2} + k^{2}, 0 < x < 1 x \geq 1$ is differentiable for all $x > 0$ . Then $\frac{8 f ^{'} ( 8 )}{f ^{'} ( \frac{1}{8} )}$ is equal to _ …
JEE Main 2023View solution →
Let $f : (- 2, 2) \to R$ be defined by
$f (x) = {x [x] (x - 1) [x], - 2 < x < 0, 0 \leq x < 2$

Where $[x]$ denotes the greatest integer function. If m and n respectively are the number of points in $(- 2, 2)$ at which $y = ∣ f (x) ∣$ is not continuous a …
JEE Main 2023View solution →
Let $[x]$ be the greatest integer $\leq x$ . Then the number of points in the interval $(- 2, 1)$ , where the function $f (x) = ∣ [x] ∣ + x - [x]$ is discontinuous is
JEE Main 2023View solution →
For the differentiable function
$f : R - {0} \to R$ , let $3 f (x) + 2 f (\frac{1}{x}) = \frac{1}{x} - 10$ , then $f (3) + f^{'} (\frac{1}{4})$ is equal to
JEE Main 2023View solution →
Let $[x]$ denote the greatest integer function and $f (x) = max {1 + x + [x], 2 + x, x + 2 [x]}, 0 \leq x \leq 2$ . Let $m$ be the number of points in $[0, 2]$ , where $f$ is not continuous and $n$ b …
JEE Main 2023View solution →
Let $f$ and $g$ be two functions defined by
$f (x) = {x + 1, ∣ x - 1∣, x < 0 x \geq 0$ and $g (x) = {x + 1, 1, x < 0 x \geq 0$ .
Then $(g \circ f) (x)$ is
JEE Main 2023View solution →