Continuous functions are locally integrable?

Question

If $K\subset\mathbb{R}$ is compact and $f:K\rightarrow\mathbb{R}$ continuous then $f\in\mathbb{L}(K)$. In other words $f$ is integrable in $K$.

So far i know that since $f$ is continuous then $f(K)$ is bounded. But I don't know how I can use this with the integral $\int_Kf$.

Answer 1

If $f$ is continuous on a compact set $K$, then there is a constant $M$ such that $|f(x)|\leq M$ for all $x\in K$. Therefore $$ \int_K|f(x)|\;dx\leq Mm(K)<\infty $$ so $f$ is integrable on $K$.