Give an example of a function $f:\mathbb{R}^2\to\mathbb{R}$ such that $f'_u(0,0)$ exists in all directions $\|u\| = 1$, but $f$ is not differentiable at $(0,0)$. You have to show that your example satisfy the above requirement.
Example of a function $f:\mathbb{R}^2\to\mathbb{R}$ not differentiable at $(0,0)$, but has a directional derivative at $(0,0)$ in all directions
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real-analysis
multivariable-calculus