As in the question title, if I can show that the hessian matrix of a function $f$ exists and is well-defined, by computing it can I then conclude that the function $f$ is twice differentiable?
For example, given an $f:\mathbb{R}^N \to \mathbb{R}$ where $\boldsymbol x\mapsto (\frac{1}{x})^{1/\alpha}$ for some $\boldsymbol x\in \mathbb{R}^N$, $\alpha>4$ where all components of $\boldsymbol x$ are bigger than $0$. Then I must show that $f$ is twice differentiable over $\mathbb{R}^N$.