IAT (IISER) 2025 | Limits, Continuity and Differentiability Question 1 | Mathematics

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=\left|x^3-3 x\right|[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Which one of the following statements is TRUE?

Every non-zero integer is a point of discontinuity of $f$

$\quad f$ is continuous at every real number

Every integer is a point of discontinuity of $f$

$f$ is continuous at every real number except for $0, \pm \sqrt{3}$

IAT (IISER) 2024

MCQ (Single Correct Answer)

-1

Let $f, g: R \rightarrow R$ be functions. If $g$ is continuous, then which one of the following cases implies that $f$ is continuous?

$g(x)=(f(x))^2$

$g(x)=|f(x)|$

$g(x)=(f(x))^3$

$g(x)=\sin (f(x))$

IAT (IISER) 2023

MCQ (Single Correct Answer)

-1

Which one of the following functions is differentiable at $x=0$ ?

$|x|$

$|x|^{\frac{1}{2}}$

$\sin |x|$

$\cos |x|$

IAT (IISER) 2020

MCQ (Single Correct Answer)

-1

If $f(x)=a x^2+b x+c$ and $f\left(\frac{1}{n}\right)=\frac{n+1}{n^2}$ for all $n \in N$, then what is the value of $\lim \limits_{x \rightarrow 0} f^{\prime}(x)$ ?

-1

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