# Continuity and Differentability of a Partial derivative

Pick the correct statement

(a) If $f(x,y)$ is continuous everywhere, then it is also differentiable everywhere.

(b) If $f(x,y)$ is continuous everywhere, then it also has partial derivatives everywhere.

(c) If $f(x,y)$ is continuous everywhere, then its partial derivatives are also continuous everywhere.

(d) If $f(x,y)$ is differentiable everywhere, then it is also continuous everywhere.

(e) If $f(x,y)$ is differentiable everywhere, then its partial derivatives are defined everywhere.

(f) If $f(x,y)$ is differentiable everywhere, then its partial derivatives are continuous everywhere.

(g) If $f(x,y)$ has partial derivatives everywhere, then it is continuous everywhere.

(h) If $f(x,y)$ has partial derivatives everywhere, then it is differentiable everywhere.

(i) If $f(x,y)$ has continuous partial derivatives everywhere, then $f(x,y)$ itself is continuous everywhere.

(j)If $f(x,y)$ has continuous partial derivatives everywhere, then it is differentiable everywhere

What I did:

I chose statements (d),(e),(f) because for a function to be differentiable everywhere its partial derivatives must exists and the function will surely be continuous everywhere. I believe (j) is also correct since I know that if all partial derivatives exists and are continuous on a neighborhood than the function is also differentiable along that neighborhood. However I seem to be missing out on some options.Also could anyone explain the three concepts of partial derivative, continuty and differentibility and how to releate them? Thanks.

• I think, d,e,i, j are correct. Partial derivatives can exist, but, that doesn't say much about continuity due to the examples where all linear paths agree but quadratics differ. I hope someone comes along with some specific counter examples soon, I must run. – James S. Cook Sep 15 '14 at 12:25