Show that the autocorrelation $R_f(x)$ converges uniformly to $R_f(x)$ - Fourier analysis

Question

we have the autocorrelation of $f(x)$:
$$R_f(x) = \frac{1}{2\pi }\int _{-\pi }^{\pi }f\left(t\right)\cdot \overline{f\left(t-x\right)}dt$$
show that the Fourier series of $R_f(x)$ converges uniformly for $R_f(x)$
It is known that $f(x)$ is partial continuous at $[-\pi, \pi]$ and periodic $2\pi$. So what I have tried:
I found $C_0$ and $C_n$ of $R_f(x)$ (complex Fourier coefficients), which gives me actually that $C_n$ and $C_0$ are equal, which means, I receive that my Fourier series of $R_f(x)$ gives me: $$\sum _{n=-\infty }^{\infty }\frac{1}{2\pi }\int _{-\pi }^{\pi }\left(\int _{-\pi }^{\pi }f\left(t\right)\cdot \overline{f\left(t-x\right)}dt\right)dx$$
Now, I am a little stuck. Since $f(x)$ is pointwise continuous, it does not mean the integral is continuous and I do not have any way to "play" with it. Any tip will be welcomed.