If $ u (x) $ is harmonic and equal to $ phi (| x |) $, is $ phi $ continuously differentiable?

I was trying to show that the radial harmonic functions in the unit's ball (in $ mathbb {R} ^ n $) are constant. For this, I suppose $ u $ It is a radial harmonic function in the unit ball and writes.
$$
u (x) = phi ( lvert x rvert)
$$

for some function $ phi $ defined in $[01)$[01)$[01)$[01)$. I got stuck trying to rigorously prove that $ phi $ It has to be differentiable.