in representing sinusoidal function of single frequency component in frequency domain it gives a single impulse at the given frequency

ok i understand it physically

but what is its mathematical proof (sinusoidal in time domain->impulse in frequency domain)

also what is the meaning of having infinite amplitude of the function at that frequency (amplitude of the impulse=infinity)........of course this is not the amplitude of the function at that frequency

and the next question .......in the laplace transform

the s that makes the F(s)=infinity is the frequency component of the system

what about the other s's ........they also give values to F(s)

but not infinity

so why are they not considered to be frequency components of the function

and what are the difference between them and between the s's that make F(s)=0???

help please because my brain is burning