There exists a function $f: \Bbb R \mapsto \Bbb R$ such that $f^{(4)}(x)$ exists for all $x$ in $\Bbb R$ but is discontinuous at $x=0$

Question

Give an example of a function $f: \Bbb R \to \Bbb R$ such that $f^{(4)}(x)=f’’’’(x)$ exists for all $x$ in $\Bbb R$ but is discontinuous at $x=0$. I tried it but I have no clue how to construct examples.

Answer 1

Hint

Let $$g(x)=\left\{ \begin{array}{lc} x^2\sin(\frac{1}{x}) & \mbox{ if } x \neq 0 \\ 0& \mbox{ if } x=0 \end{array} \right. $$

Then $g$ is differentiable everywhere, but $g'$ is discontinuous at $x=0$.

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Answer 2

Let consider

$$f(x)=\left\{ \begin{array}{lc} 0 & \mbox{ if } x < 0 \\ x^4& \mbox{ if } x\ge0 \end{array} \right.$$