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

Question

I was trying to show that radial harmonic functions on the unit ball (in $\mathbb{R}^n$) are constant. To this end, I suppose that $u$ is a radial harmonic function on the unit ball and write $$ u(x) = \phi(\lvert x\rvert) $$ for some function $\phi$ defined on $[0, 1)$. I got stuck when trying to rigorously prove that $\phi$ has to be differentiable.

Answer 1

Let $s$ be any vector with norm $1$. Then $\phi(t)=u(ts)$, thus $\phi$ is continuously differentiable, since $u$ is.

Answer 2

Hint: Use the MVP to see $u(0) = \phi(r)$ for $0\le r <1.$