Second Derivative Test: Can we relax hypothesis that $f$ is twice differentiable on neighborhood? (Counterexample)

Question

Second Derivative Test for Extrema:

Let $f:\mathbb R \to \mathbb R$ be a function that is twice-differentiable on $(c-\varepsilon,c+\varepsilon)$ for some $\varepsilon >0$. Suppose $f^{\prime}\left(c\right)=0$.

1. If $f^{\prime\prime}\left(c\right)<0$, then $c$ is a strict local maximum of $f$.

2. If $f^{\prime\prime}\left(c\right)>0$, then $c$ is a strict local minimum of $f$.

Can we delete "twice-" in the first sentence and just assume $f^{\prime\prime}(c)$ exists?

I'm guessing not but what might a counterexample be?