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Let $f$ be the function defined on the real line by

$f(x) =$ \begin{cases} 2x^2, & \text{if $x \in \mathbb{Q}$} \\ -3x^2, & \text{if $x \notin \mathbb{Q}$} \\ \end{cases}

Then which for the following is true?

A. $f$ is not continuous and not differentiable everywhere

B. $f$ is continuous only at $x=0$ and not differentiable everywhere

C. $f$ is continuous and differentiable only at $x=0$

D. $f$ is continuous and not differentiable everywhere

E. $f$ is continuously differentiable everywhere

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3 Answers

Use the sequence of characterisation of continuity, together with the familiar property of reals: to every real number, there are two sequences, one of rationals and other of irrationals converging to it (in $\mathbf{R}$).

The differentiability can be investigated based on similar ideas.

BTW, the answer is (C)

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I think a picture helps: ${}{}{}$

enter image description here

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Answer c is correct. For derivability at 0 take the definition. Recall that a derivable function at a point is also continuous.

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