From Schaum's Outlines of Signals & Systems:
Let's work with continuous-time signals.
Let $T$ be a linear time-invariant system (LTI).
Input $x(t)$ can be expressed as $x(t) = \int_{-\infty}^{\infty} x(\tau) \delta(t-\tau) d \tau$.
Then output $y(t)$ can be expressed as $y(t) = \int_{-\infty}^{\infty} x(\tau) T\{\delta(t - \tau)\}d\tau$
I'm not sure how they arrived at the last bolded statement without showing how. My first thought is to break down the integral into a sum, but not sure if you can do that when working with $\delta$, and it seems like a non-elegant approach. Please lucidate on a better approach. Thanks.