I would just like someone to check this proof for me. I am just learning this material so I would like to make sure I am doing the proofs properly.

Question: Show that if and and if the partial derivatives exist and are bounded in a neighborhood of , then f is continuous at

Proof: We have to show that as where is a real number and is any basis vector for .

Let be an upper bound for all the partial derivatives of f, in some neighborhood of . Observe that and so as desired. QED