Why the definition of harmonic function needs twice " continuously " differentiable?

Question

Let $u$ denote a twice continuously differentiable function , then $u$ is harmonic if it satisfies Laplace's equation : $$\Delta u=\sum_{j=1}^d \frac{\partial^2u}{\partial x_j^2}=0$$
However , if $f$ is a twice differentiable function which is not twice continuous differentiable , can $f$ satisfies that Laplace's equation ?