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 $f (x) = [x^{2} - x] + ∣ - x + [x] ∣$ , where $x \in R$ and [t] denotes the greatest integer less than or equal to t. Then, f is
JEE Main 2023View solution →
Let $f (x) = {x^{2} sin (\frac{1}{x}) 0, x \neq = 0, x = 0$ ; Then at $x = 0$
JEE Main 2023View solution →
Let $f : R \to R$ be a differentiable function that satisfies the relation $f (x + y) = f (x) + f (y) - 1, \forall x$ , $y \in R$ . If $f^{'} (0) = 2$ , then $∣ f (- 2) ∣$ is equal …
JEE Main 2023View solution →
If the function
$f (x) = ⎩ ⎨ ⎧ (1 + ∣ cos x ∣) \frac{λ}{∣ c o s x ∣}, μ, e^{\frac{c o t 6 x}{c o t 4 x}} 0 < x < \frac{π}{2} x = \frac{π}{2} is continuous at, \frac{π}{2} < x < π$
$x = \frac{π}{2}$ , th …
JEE Main 2023View solution →
Let the function
$f (x) = {\frac{l o g _{e} ( 1 + 5 x ) - l o g _{e} ( 1 + α x )}{x} 10; if x \neq = 0; if x = 0$
be continuous at $x = 0$ .
Then $α$ is equal to:
JEE Main 2022View solution →
If $[t]$ denotes the greatest integer $\leq t$ , then number of points, at which the function $f (x) = 4∣2 x + 3∣ + 9 [x + \frac{1}{2}] - 12 [x + 20]$ is not differentiable in the open interval $(- 20, 20)$ , is ____.
JEE Main 2022View solution →
The function $f : R \to R$ defined by
$f (x) = lim_{n \to \infty} \frac{c o s ( 2 π x ) - x ^{2 n} s i n ( x - 1 )}{1 + x ^{2 n + 1} - x ^{2 n}}$ is
continuous for all x in
JEE Main 2022View solution →
Let $f : R \to R$ be a function defined as $f (x) = a sin (\frac{π [ x ]}{2}) + [2 - x], a \in R$ , where $[t]$ is the greatest integer less than or equal to $t$ . If $lim_{x \to - 1} f (x)$ exists, then the value of …
JEE Main 2022View solution →
Let a function $f : R \to R$ be defined as:
$f (x) = {\int_{0}^{x} (5 - ∣ t - 3∣) d t, x^{2} + b x, x > 4 x \leq 4$
where $b \in R$ . If $f$ is continuous at $x = 4$ , then which of the following statements is NOT true?
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Let $f (x) = {4 x^{2} - 8 x + 5, if 8 x^{2} - 6 x + 1 \geq 0 [4 x^{2} - 8 x + 5], if 8 x^{2} - 6 x + 1 < 0$ ,
where $[α]$ denotes the greatest integer less than or equal to $α$ . Then the number of points i …
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If for $p \neq = q \neq = 0$ , then function
$f (x) = \frac{7 p ( 729 + x ) - 3}{3 729 + q x - 9}$ is continuous at $x = 0$ , then:
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Let $f (x) = 2 + ∣ x ∣ - ∣ x - 1∣ + ∣ x + 1∣, x \in R$ . Consider
(Sl): …
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For the curve $C : (x^{2} + y^{2} - 3) + (x^{2} - y^{2} - 1)^{5} = 0$ , the value of $3 y^{'} - y^{3} y^{''}$ , at the point $(α, α)$ , $α > 0$ , on $C$ , is equal to ____.
JEE Main 2022View solution →
If the function
$f (x) = {\frac{l o g _{e} ( 1 - x + x ^{2} ) + l o g _{e} ( 1 + x + x ^{2} )}{s e c x - c o s x}, k, x \in (\frac{- π}{2}, \frac{π}{2}) - {0} x = 0$ is continuous at $x = 0$ , then k is equal to :
JEE Main 2022View solution →
If $f (x) = {x + a, ∣ x - 4∣, x \leq 0 x > 0$ and
$g (x) = {x + 1 (x - 4)^{2} + b, x < 0 x \geq 0$
are continuous on R , then (gof) (2) $+ (fog) (- 2)$ is equal to :
JEE Main 2022View solution →
Let $f, g : R \to R$ be functions defined by
$f (x) = {[x] ∣1 - x ∣,, x < 0 x \geq 0$ and
$g (x) = {e^{x} - x (x - 1)^{2} - 1 x < 0, x \geq 0$
where $[x]$ deno …
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Let $f : R \to R$ be defined as …
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Let $f (x) = min {1, 1 + x sin x}, 0 \leq x \leq 2 π$ . If $m$ is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair ( $m, n$ ) is equal to
JEE Main 2022View solution →
Let $f : R \to R$ be a function defined by :
$f (x) = ⎩ ⎨ ⎧ max {t^{3} - 3 t}; x \leq 2 t \leq x x^{2} + 2 x - 6; 2 < x < 3 [x - 3] + 9; 3 \leq x \leq 5 2 x + 1; x > 5 ⎭ ⎬ ⎫$
Where $[t]$ is the greatest …
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Let $f, g : R \to R$ be two real valued functions defined as $f (x) = {- ∣ x + 3∣ e^{x}, x < 0, x \geq 0$ and …
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The number of points, where the function $f : R \to R, f (x) = ∣ x - 1∣ cos ∣ x - 2∣ sin ∣ x - 1∣ + (x - 3) x^{2} - 5 x + 4$ , is NOT differentiable, is :
JEE Main 2022View solution →
The number of points where the function
$f (x) = ⎩ ⎨ ⎧ 2 x^{2} - 3 x - 7 [4 x^{2} - 1] ∣ x + 1∣ + ∣ x - 2∣ if if if x \leq - 1 - 1 < x < 1 x \geq 1$
$[t]$ denotes the greatest intege …
JEE Main 2022View solution →
Let $f (x) = {max {2 x, 3 [∣ x ∣]} 1, ∣ x ∣ < 1, otherwise$
where $[t]$ denotes greatest integer $\leq t$ . If m is the number of points where $f$ is not continuous and n is the number of points where $f$ is not differentiable, then the ordered …
JEE Main 2022View solution →
Let $f (x) = [2 x^{2} + 1]$ and $g (x) = {2 x - 3, 2 x + 3, x < 0 x \geq 0$ , where $[t]$ is the greatest integer $\leq t$ . Then, in the open interval $(- 1, 1)$ , the number of points where fog is discontinuous is equal to
JEE Main 2022View solution →
If $y = y (x)$ is an implicit function of $x$ such that $lo g_{e} (x + y) = 4 x y$ , then $\frac{d ^{2} y}{d x ^{2}}$ at $x = 0$ is equal to ____.
JEE Main 2021View solution →
Let $a, b \in R, b \neq = 0$ . Define a function $f (x) = {a sin \frac{π}{2} (x - 1), \frac{t a n 2 x - s i n 2 x}{b x ^{3}}, for x \leq 0 for x > 0$ If $f$ is continuous at $x = 0$ , then $10 - ab$ is equal to ____.
JEE Main 2021View solution →
If $f : R \to R$ is a function defined by $f (x) = [x - 1] cos (\frac{2 x - 1}{2}) π$ , where [.] denotes the greatest integer function, then $f$ is:
JEE Main 2021View solution →
Let $[t]$ denote the greatest integer less than or equal to $t$ .
Let $f (x) = x - [x]$ , $g (x) = 1 - x + [x]$ , and $h (x) = min {f (x), g (x)}$ , …
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Let a function $f : R \to R$ be defined as $f (x) = ⎩ ⎨ ⎧ sin x - e^{x} a + [- x] 2 x - b if if if x \leq 0 0 < x < 1 x \geq 1$
where $[x]$ is the greatest integer less than or equal to $x$ . If …
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Let a function $g : [0, 4] \to R$ be defined as $g (x) = {max_{0 \leq t \leq x} {t^{3} - 6 t^{2} + 9 t - 3} 4 - x, 0 \leq x \leq 3, 3 < x \leq 4$ , then the number of points in the interval $(0, 4)$ where …
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The number of points, at which the function $f (x) = ∣2 x + 1∣ - 3∣ x + 2∣ + x^{2} + x - 2, x \in R$ is not differentiable, is
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Let $f : R \to R$ be defined as $f (x) = {\frac{x ^{3}}{( 1 - c o s 2 x ) ^{2}} lo g_{e} (\frac{1 + 2 x e ^{- 2 x}}{( 1 - x e ^{- x} ) ^{2}}) α, x \neq = 0, x = 0$ . If $f$ is continuous at $x = 0$ , then $α$ i …
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Let $f : R \to R$ be a function defined as $f (x) = {3 (1 - \frac{∣ x ∣}{2}) 0 if ∣ x ∣ \leq 2 if ∣ x ∣ > 2$

Let $g : R \to R$ be given by $g (x) = f (x + 2) - f (x - 2)$ . If $n$ and …
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Let f be any function defined on $R$ and let it satisfy the condition :
$∣ f (x) - f (y) ∣ \leq (x - y)^{2}, \forall (x, y) \in R$

If $f (0) = 1$ , then :
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Let $f : R \to R$ be defined as
$f (x) = ⎩ ⎨ ⎧ 2 sin (- \frac{π x}{2}), a x^{2} + x + b, sin (π x), if x < - 1 if - 1 \leq x \leq 1 if x > 1$

If $f (x)$ is continuous on $R$ , …
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The function $f (x) = x^{2} - 2 x - 3 \cdot e^{∣ 9 x^{2} - 12 x + 4 ∣}$ is not differentiable at exactly:
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If the function $f (x) = ⎩ ⎨ ⎧ \frac{1}{x} lo g_{x} (\frac{1 + \frac{x}{a}}{1 - \frac{x}{b}}), x < 0 k, x = 0 \frac{c o s ^{2} x - s i n ^{2} x - 1}{x ^{2} + 1 - 1}, x > 0$
is continuous at $x = 0$ , then …
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Let $f : (- \frac{π}{4}, \frac{π}{4}) \to R$ be defined as
$f (x) = ⎩ ⎨ ⎧ (1 + ∣ sin x ∣)^{\frac{3 a}{∣ s i n x ∣}} b e^{c o t 4 x / c o t 2 x},,, - \frac{π}{4} < x < 0 x = 0 0 < x < \frac{π}{4}$

If $f$ is continuo …
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Let $f : [0, 3] \to R$ be defined by $f (x) = min {x - [x], 1 + [x] - x}$
Where $[x]$ is the greatest integer less than or equal to $x$ .
Let $P$ denote the set containing all $x \in [0, 3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x \in (0, 3)$ where $f$ is n …
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If the function $f (x) = \frac{c o s ( s i n x ) - c o s x}{x ^{4}}$ is continuous at each point in its domain and $f (0) = \frac{1}{k}$ , then k is ____
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Let $f : R \to R$ be defined as
$f (x) = ⎩ ⎨ ⎧ \frac{λ ∣ x ^{2} - 5 x + 6 ∣}{μ ( 5 x - x ^{2} - 6 )} e^{\frac{t a n ( x - 2 )}{x - [ x ]}} μ, x < 2, x > 2, x = 2$

where $[x]$ is the greatest integ …
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Let $f : [0, \infty) \to [0, \infty)$ be defined as
$f (x) = \int_{0}^{x} [y] dy$
where $[x]$ is the greatest integer less than or equal to $x$ . Which of the following is true?
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If $f (x) = {\int_{0}^{x} (5 + ∣1 - t ∣) dt, 5 x + 1, x > 2 x \leq 2$ , then
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Consider the function $f (x) = \frac{P ( x )}{s i n ( x - 2 )}, x \neq = 2$
$= 7, x = 2$

where $P (x)$ is a polynomial such that $P^{''} (x)$ is always a constant and $P (3) = 9$ . If $f (x)$ is continuous at $x = 2$ , then $P (5)$ is equal to...... .
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Let $f : [0, \infty) \to [0, 3]$ be a function defined by $f (x) = {max {sin t : 0 \leq t \leq x}, 2 + cos x, 0 \leq x \leq π x > π$
Then which of the following is true?
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Let $[t]$ denote the greatest integer $\leq t$ . The number of points where the function $f (x) = [x] x^{2} - 1 + sin (\frac{π}{[ x ] + 3}) - [x + 1], x \in (- 2, 2)$ is not continuous is
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Let the functions $f : R \to R$ and $g : R \to R$ be defined as :
$f (x) = {x + 2, x^{2}, x < 0 x \geq 0$ and $g (x) = {x^{3}, 3 x - 2, x < 1 x \geq 1$
Then, the number of poin …
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Let $α \in R$ be such that the function $f (x) = {\frac{c o s ^{- 1} ( 1 - { x } ^{2} ) s i n ^{- 1} ( 1 - { x })}{{ x } - { x } ^{3}} α, x \neq = 0, x = 0$ is continuous at $x = 0$ , where ${x} = x - [x], [x]$ is the greatest inte …
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Let $f : S \to S$ where $S = (0, \infty)$ be a twice differentiable function such that $f (x + 1) = x f (x)$ . If $g : S \to R$ be defined as $g (x) = lo g_{e} f (x)$ , then the value of $∣ g^{''} (5) - g^{''} (1) ∣$ is equal to :
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Let $f : R \to R$ and $g : R \to R$ be defined as
$f (x) = {x + a, ∣ x - 1∣, x < 0 x \geq 0$ and $g (x) = {x + 1, (x - 1)^{2} + b, x < 0 x \geq 0$ , where $a, b$ are non-negative rea …
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