This question may seem slightly unnecessary but I'm really curious about the answer. For a system (LTI ones at least not sure about the rest) the output can be described as some function of the input: $$y(t) = f(x(t))$$ So surely the impulse response of a system can be thought of the same way: $$h(t) = f(\delta (t))$$ Now let's say we have the impulse response: $$h(t) = e^{-t}u(t) = f(\delta (t))$$ This could be the sound of a clap decaying in a room or something.
What I'm interested in is understanding what kind of function $f$ is able turn the delta function, which only exists for the briefest moment, into a decaying function that exists for a period of time.
When I think of it in terms of the decaying sound in a room, it makes perfect sense that the response would be a decaying function because its just the sound bouncing off the walls creating an echo that decays away, but I can't see mathematically how any function could take the delta function and somehow create this kind of a response.
I'm not sure if this question has a definite answer, I'm sorry if it doesn't, I just really hope it does. The delta function always seems like some math tool that was created because it's useful as opposed to representing real life.
Thanks, Richard
By the definition of impulse response, $f()$ is the convolution operator (as shown in @John Doe's comments)
$$f(x(t)) = h(t) * x(t) $$
And now note that
$$f(\delta(t)) = h(t) * \delta(t) = h(t)$$