# $\phi$ is a step function. Prove that $|\phi|$ is a step function

Let $$\phi :[a,b] \rightarrow \Bbb R$$ be a step function.

I have to prove that $$|\phi|$$ is a step function. Here's how I prove it:

Let $$P$$ be a partition $$P=\{p_0,...,p_k\}$$ on $$[a,b]$$, compatible with $$\phi$$. Let $$\phi_i$$ be the value that $$\phi$$ takes on each interval $$(p_{i-1},p_i)$$, for $$1\leq i \leq k$$. We see that $$|\phi|$$ takes values $$|\phi_i|$$ on the intervals $$(p_i,p_{i-1})$$, hence by definition, $$|\phi|$$ is a step function.

Is my proof correct? It seems too short to me, but I think there's nothing more to it really. Any feedback is appreciated

• This proof is correct. – Boka Peer Mar 25 '20 at 13:24