real analysis - Evaluate $limlimits_{ntoinfty}nintlimits_0^1 f(x)e^{-nx}mathrm dx$ where $,f$ is bounded in $mathbb{R}^+cup{0}$

Evaluate $\lim\limits_{n\to\infty}n\int\limits_0^1 f(x)e^{-nx}\mathrm dx$ where $\,f$ is bounded in $\mathbb{R}^+\cup\{0\}$.

My problem is that I think there's missing information about $f$, e.g. some kind of continuity on $0$. Because if we change the variable of integration for $\frac xn$ the integral is equal to
$$\lim\limits_{n\to\infty}\int\limits_0^n f\left(\frac xn\right)e^{-x}\mathrm dx$$
And that can be dominated by $Me^{-x}$, where $|f|\leq M$. But the convergence is to a function non continue (necessary).

Am I wrong?