Let $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\mathop \smallint \limits_0^x \left\{ {5 + \left| {1 - t} \right|} \right\}dt,{\rm{if}}\,\,x > 2}\\ {5x + 1,{\rm{if}}\,\,x \le 2} \end{array}} \right.$

$f\left( x \right)$ is continuous at $x = 2$

$f\left( x \right)$ is everywhere differentiable

The right derivative of $f\left( x \right)at\,\,{\rm{ }}x = 2$ does not exist

JEE MATHEMATICS MCQ: CONTINUITY

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Analyzing +4 | -1

Let

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\int \limits_0^x \left\{ {5 + \left| {1 - t} \right|} \right\}dt,\mathrm{if}\,\,x > 2}\\ {5x + 1,\mathrm{if}\,\,x \le 2} \end{array}} \right.

View answer and explanation

Correct option: B

f\left( x \right)

is continuous but not differentiable at

x = 2

Option B is correct because

f(x)

is continuous at

x=2

but the left-hand derivative and right-hand derivative are not equal, making it non-differentiable.

Practice All Chapter MCQs →

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {{{\left\{ {1 + \left| {\sin x} \right|} \right\}}^{\frac{a}{{\left| {\sin x} \right|}}}}, - \frac{\pi }{6} < x < 0}\\ {b,x = 0}\\ {{e^{\frac{{\tan 2x}}{{\tan 3x}}}},0 < x < \frac{\pi }{6}} \end{array}} \right.

x = 0

f(x)=\left\{\begin{array}{rr}1+\cos ^{-1}\{\cot x\}, & x<\frac{\pi}{2} \\ \pi[x]+1, & x \geq \frac{\pi}{2}\end{array}\right.

\{ \}

f:R \to R

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\frac{{\cos 3x - \cos x}}{{{x^2}}},\mathrm{for}\,\,x \ne 0}\\ {\lambda ,\mathrm{for}\,\,x = 0} \end{array}} \right.

f

x = 0

\lambda

f

f(x) f(y)=

f(x)+f(y)+f(x y)-2

x, y \in {R}

f(2)=5

\lim \limits_{{x \rightarrow 4}} f(x)

\mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}\frac{x^2+3 x-10}{x^2+2 x-15} \text {, when } x \neq-5 \\ a, \quad \text { when } x=-5\end{array}\right.

x=-5

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {x\sin \frac{1}{x},x \ne 0}\\ {k,x = 0} \end{array}} \right.

x = 0

k