Integral FunctionProperty Intermediate

Problem - 4554

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that

$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$

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