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Let $f:\mathbb{R}^2\to \mathbb{R}$ be any function. Then we know the following:

  1. Differentiability implies existence of partial derivatives and continuity
  2. Existence of partial derivatives does not imply continuity and hence not differnetiability.
  3. Continuity does not imply differentiability.

But now my question is "Does continuity of $f$ implies existence of partial derivatives?"

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No. Take $f(x,y)=|x|$. It is continuous, but $\frac{\partial f}{\partial x}(0,0)$ doesn't exist.

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Not even nicely true for single-variable functions. See Weierstrass function: https://en.wikipedia.org/wiki/Weierstrass_function

Continuous everywhere, differentiable nowhere.

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