Differentiability at a Point when all Partial Derivatives are Discontinuous at that point

Suppose $$f: R^n\to R^m$$ n > 1 s.t. all the partial derivatives are discontinuous at point a. Is the function differentiable at a? I would think not however, I know of an example in R that works, so I am wondering how this works with the linear operator of differentiability.

Hint: Let $$f$$ be your example in one variable. Think about $$f(x)+f(y).$$