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Let $f(x)= |x|^{3/2}, x \in \mathbb{R}$. Then

1. $f$ is uniformly continuous.
2. $f$ is continuous, but not differentiable at $x=0$.
3. $f$ is differentiable and $f '$ is continuous.
4. $f$ is differentiable, but $f '$ is discontinuous at $x=0$.
asked in Calculus | 152 views

f is differential & first derivative of f is discontinuous at x =0

The graph for $f(x) = |x|^{\frac{3}{2}}$ looks like :

And the plot of its derivative is :

Clearly : option D

f(x) is not continuous because the domain is all R, but the plot is only possible for positive real numbers.