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.