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Consider a function $f:\mathbb{R}^l\rightarrow \mathbb{R}$ continuously differentiable at $x_0$. This implies that

(1) the function is differentiable on a neighbourhood of $x_0$ which means that the function is continuous on a neighbourhood of $x_0$

(2) the partial derivatives are all well-defined on a neighbourhood of $x_0$.

Are (1) and (2) correct?

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The definition of 'continuous differentiable' is usually applied to functions defined on an open set and differentiability on that open set is assumed to be true as a prerequisite. So the first question would be what continuous differentialbility at some point is supposed to mean, and the obvious definition would be to say it's continuously differentiable in some neighbourhood of that point (I guess this can be weakened, but I doubt that this is the point here).

So, in a way, yes, (1) is implied, but I'd prefer to say it's true by assumption. (2) is, then, in fact, also true.

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