How to prove a constant function is $C^{\infty}$ Let $U\subset R^m$ be an open set and $f:U\to\mathbb R^n$ a constant function. My question is really simple. I know that this function is $C^{\infty}$, but I'm having troubles to prove it formally. What I know is its derivative is zero and $f \in C^k$ iff $f'\in C^{k-1}$. I don't know how to manage all this information to prove $f\in C^k$ for every $k=0,1\ldots$
I've already proved $f\in C^0$ and $f\in C^1$. My problem is to prove the induction part: $f\in C^k\implies f\in C^{k+1}$

Let's strengthen your induction hypothesis. We prove $f \in \mathcal{C}^{k}$ and $d^kf = 0$ for all $k \ge 1$.


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*$k = 1$: you should prove $df=0$, using the definition of differential.

*If $f \in \mathcal{C^{k-1}}$ and $d^{k-1}f = 0$, you should prove $f \in \mathcal{C^{k}}$ and  $d^{k}f = 0$. Once again apply the definition to the hypothesis.

