For a function of a single variable, I can check if the function is differentiable by simply taking the limit definition of a derivative and if the limit exists, then the function is differentiable at that point. Differentiability also implies continuity at this level.(adsbygoogle = window.adsbygoogle || []).push({});

Now, for a function of say, f(x,y), a function of just two variables, if I want to know if f is differentiable at a point c things seem more complicated. I know I can take the limit definition of partial derivatives and say check if the limit for the partial with respect to x exists at c, and if the limit for the partial with respect to y exists at c. But it seems this won't be sufficient because I know that there are infinitely many directional derivatives. Wouldn't all of those directional derivatives need to exist at c for the function to be differentiable at c? Is there some general way to check this withi a limit definition, say by looking at the gradient? (I am asking about the gradient because I know I can find a directional derivative in any direction u by taking the dot product of the gradient at c and the unit vector in the direction of u)

Also, if every single partial derivative exists, can we then say the function is continuous as a result of being differentiable?

(A side question: what if on the single variable level I have a piecewise function like:

f(x) = {(x^2-4x+4)/(x-2) if x =/= 2,

0 if x x==2}

This function would be differentiable at x = 2 right? I'm sorry but the piecewise functions just confuse me when it comes to these things)

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# Differentiability of a multi-variable function

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