Filter response to sinusoidal input signal
The fact that a sinusoidal signal's amplitude is multiplied by the magnitude of the system's frequency response (evaluated at the input frequency) is a general property of LTI systems, regardless of whether they are continuous-time or discrete-time systems.

This is a consequence of the complex exponential being an eigenfunction of LTI systems, and the value of the frequency response evaluated at the input frequency is the corresponding eigenvalue.

For more details on eigenfunctions of LTI system check out this question and its answers. And I'm sure that there's even more information here on this site.