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Let $f$ be a real-valued function of a real variable defined as $f(x)=x^2$ for $x \geq 0$, and $f(x)=-x^2$ for $x<0$. Which one of the following statements is true?

- $f(x)$ is discontinuous at $x=0$
- $f(x)$ is continuous but not differentiable at $x=0$
- $f(x)$ is differentiable but its first derivative is not continuous at $x=0$
- $f(x)$ is differentiable but its first derivative is not differentiable at $x=0$