Let us define the Fourier transform in the signal processing convention:
\begin{align} S(f)&=\int_{-\infty}^{+\infty}s(t)e^{-i2\pi f t}dt \\ s(t)&=\int_{-\infty}^{+\infty}S(f)e^{i2\pi f t}df \quad . \end{align}
First, let $s=\delta_0$ be the Dirac delta's distribution. It is the inverse Fourier transform of $S(f)=1$: \begin{align} \delta(t)&=\int_{-\infty}^{+\infty}e^{i2\pi f t}df \quad . \end{align}
Then, consider a bivariate function $S(t,f)$ so that its inverse Fourier transform with respect to $f$ is the Dirac delta: \begin{align} \int_{-\infty}^{+\infty}S(t,f)e^{i2\pi f t}df&=\delta(t) \quad . \end{align}
My question: