I'm reading a rather informal text on (continuous) Linear Time Invariant (LTI) systems. It is just said to be a "black box" that transforms an input signal $x(t)$ into an output signal $y(t)=(\mathcal{H}x)(t)$ which in addition satisfies
$\alpha x_1(t) + \beta x_2(t) \longrightarrow \alpha y_1(t) + \beta y_2(t)$
$x(t-a) \longrightarrow y(t-a)$
It is continuous in the sense that "small changes of $x$ results in small changes of $y$".
I hope the above notation is clear without any further explanations.
Then it is said that such a system can always be written in the convolution form
$$
y(t) = (h\ast x)(t)
$$
for some $h$ that happens to be the result of taking the input signal $x=\delta $. I'm just wondering how to prove this claim? The only way I can think of would be to write
$$
x(t) = \int \limits _{-\infty } ^{\infty } \delta (t-s)x(s)\,ds
$$
and apply $\mathcal{H}$ inside the integral. But is that allowed? Or do I have to treat the integral as a Riemann sum, use the linearity and then somehow appeal to the continuity property?