How to obtain the CTFT of a unit impulse formula in terms of f in hertz (from the formula in terms of $ \omega $)
Recall:
$ \mathcal{X}(\omega)=1 $
To obtain X(f), use the substitution
$ \omega= 2 \pi f $.
More specifically
$ X(f)=\mathcal{X}(2\pi f)=1\ $
