Below is the problem that I described in my title and a proof for it below. Before looking at this solution I was trying to prove it by showing that the partials are continuous, which would imply that $f$ is differentiable and hence continuous. I had trouble using the bounded property to show that $|\partial_xf(x,y)-\partial_xf(a,b)| \leq |(x,y)-(a,b)|$. Is it possible to prove this inequality using the bounded property? and is it at all possible to prove differentiability of $f$ as the method of proving continuity? I would also be interested to see other proofs of the coninuity of $f$ such as by the way of the open set condition, or using sequential convergence.
Solution
