I'm trying to answer this problem: Consider a real function f, bandlimited by frequency $\omega$, which satisfy $$\int_{-\infty}^\infty f(x)^2dx=c.$$ (For pure mathematicians: "bandlimited" means that it Fourier transform is supported on $[-\omega,\omega]$.)

Is the derivative of this function limited in absolute value? That is, is there an expression $A(\omega,c)$ for which $|f'(x)|<A(\omega,c)$ for all x?

Thank for the help!