Suppose $f(x) = g(x) + h(x)$ where $g(x)$ is known to be non-differentiable. When $h(x) \ne -g(x)$ is some other function (differentiable or not), can $f(x)$ ever be differentiable? We can assume $f, g, h$ are real valued functions.
edit: I should have specified that $h(x)$ is not some function that cancels out $g(x)$ . Bill Dubuque's answer is intriguing.