How to obtain the CTFT of a shifted unit impulse in terms of f in hertz (from the formula in terms of $ \omega $)
$ x(t)=\delta (t-t_0) $
$ \mathcal{X}(\omega )=e^{-iwt_0} $
To obtain X(f), use the substitution
$ \omega= 2 \pi f $.
More specifically
$ \begin{align} X(f)&=\mathcal{X}(2\pi f)=e^{-i2\pi ft_0} \end{align} $
