I am looking at the description of LTI systems in the time domain.

Intuitively, I'd have guessed it would be the composition of the input function and some "system function". $$ y(t) = f(x(t)) = (f\circ x)(t)$$ Where $x(t)$ is the input, $y(t)$ output and $f(x)$ a "system function".

Why is it not that way? Could such a "system function" be found for, say, an R-C-Circuit?

The actual output function y(t), is defined as $$ y(t) = (h * x)(t) $$ Where $h(t)$ is the response to a dirac impulse. This is hard to grasp for me. Why is it so? I have looked at various explanations, drawings of rectangles becoming infinitely narrow, which I sort of understood, but it is still "hard to grasp"! I am looking for a simple explanation in one or two sentences here.

instantaneousbehaviour of $x$—one can determine the response *right now* by knowing only the input *right now*—whereas the former allows the behaviour of $x$ at all (past) times to have an effect on the present response. The latter behaviour is what one expects out of a real-world circuit. Of course, this doesn't say why convolution (as opposed to any other integral transform—they all exhibit this sort of behaviour) is the ‘right’ – L Spice Jun 21 '11 at 20:47