TITLE: What is the difference between a function and a distribution?
QUESTION [24 upvotes]: I remember there was a tongue-in-cheek rule in mathematical analysis saying that to obtain the Fourier transform of a function $f(t)$, it is enough to get its Laplace transform $F(s)$, and replace $s$ by $j\omega$. Because their formula is pretty much the same except for the variable of integration.
And I know that this is not necessarily true (that's why I used the term tongue-in-cheek) e.g. take $f(t)=1$ to see the obvious difference.
I read somewhere that although their formula is somehow similar, their result is not necessarily similar because Laplace transform is a function and Fourier transform is a distribution. For example, the Dirac delta, $\delta(\omega)$ is a distribution and not a function.
So this led me to wonder:


What is the difference between a function and a distribution? (preferably in layman's terms)
Why the Laplace transform is a function and Fourier transform is a distribution? I mean, they are both infinite integrals. So what am I missing?

REPLY [6 votes]: Short answer by analogy: distributions are to functions as real numbers are to rational ones. The real number set can be defined as all rational numbers with the limits of all convergent infinite sequences of rational numbers added in, so distributions would be the set of all functions plus all infinite sequences of functions that "converge" in some sense. The Dirac delta function, which is the classic example of a distribution, has many equivalent definitions just like there are many different ways to calculate $\pi$ or $\operatorname{e}$. One frequently used definition is:
$$\delta(x) = \lim_{\sigma\rightarrow 0} \frac{1}{\sigma\sqrt{2\pi}} \operatorname{e}^{-\frac{1}{2} \left(\frac{x}{\sigma}\right)^2}.$$
Formally, just like how every real calculation done is performed on rational numbers, the limit is only taken after it's argument has been used in an integral. As a practical matter, though, just like taking integrals and derivatives, we can short circuit that formal process in most cases.