Easy. Let f be a continuous real function that is not differentiable somewhere. Let H be the truth value of the Riemann hypothesis. Define g(x) as "f(x) if not H else 0". Boom.
Reminds me of a homework exercise in theory of computation back in my uni days. Is the function "f(x) is 0 if H else 1" computable? Of course it is. It is either constant 0 or constant 1. We don't know which at this point, but f is a constant function in either case, so it is surely computable.