$f \in L_2$ bandlimited implies $f$ equal to continous function a.e. (without using Parley-Wiener)

Question

I was wondering, if my proof is right as I didn't find any similar statements in books or the internet without using the Parley-Wiener-Theorem:
If we have $f \in L_2(\mathbb{R})$, bandlimited (i.e. supp$(\mathcal{F} f) \subseteq [-\lambda, \lambda]$ ), then $\mathcal{F} f$ is in $L_1(\mathbb{R}) \cap L_2(\mathbb{R})$ (in $L_2$ per Definition of the $L_2$-Fourier Transform and in $L_1$ because of bounded support). The invers $L_2$-Transform $\mathcal{F}^{-1}\mathcal{F} f=f$ a.e therefore coincides with the invers $L_1$-Transform a.e. But the propertys of the (invers) $L_1$-Transform imply $\mathcal{F}^{-1}\mathcal{F} f\in L^{\infty}$ uniformly continous. We can conclude, that $f$ is a.e. equal to an uniformly continous function.