Time domain and frequency domain analysis equivalence
now, even a steady-state sinusoid can be thought of, in the limit, as a sum of weighted pulses, each with a beginning and some kinda end.  how do all of these "transient" signals add up to a steady-state sinusoid?  but they do:

$$\begin{align}
x(t) &amp;= \lim_{T \to 0} \sum_{n=-\infty}^{+\infty} x(nT) \cdot \operatorname{rect}\left( \frac{t-nT}{T} \right) \\
 &amp;= \lim_{T \to 0} \sum_{n=-\infty}^{+\infty} x(nT) \cdot \frac{1}{T}\operatorname{rect}\left( \frac{t-nT}{T} \right) \cdot T \\
 &amp;= \int\limits_{-\infty}^{\infty} x(u) \cdot \delta(t-u) \cdot du
 \end{align}$$

where $ \operatorname{rect}(u) \triangleq \begin{cases} 1 \quad |u|&lt;\tfrac12 \\ 0 \quad |u| &gt; \tfrac12  \end{cases} $

The Fourier integral models the entire time function as an infinitely dense and infinitely large set of sinusoids.  in a similar manner, those steady-state functions can add to something with a transient.

$$ x(t) = \int\limits_{-\infty}^{+\infty} X(f) \, e^{j2\pi ft} \ df $$