# [Math] Proof verification (to show that every continuous function is measurable)

lebesgue-integrallebesgue-measuremeasure-theoryproof-verificationreal-analysis

The problem is:

Show that every continuous function $f: \mathbb{R}^d \to [0,+\infty]$ is measurable.

I'm trying to use the following open set formulation of measurable function:

For all open set $U \subset [0,+\infty)$, $f^{-1}(U)$ is measurable.

Now if $f$ is continuous, then for all open set $U \subset [0,+\infty)$, $f^{-1}(U)$ is open. We know that all open sets are measurable. Hence the open set formulation is satisfied and we see that $f$ is measurable.

The proof seems too straightforward and I'm a bit suspicious about holes in it. Any help, confirmation or correction, would be greatly appreciated.