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I do apologize if this question is a bit vague, but I shall try to be as clear as possible.

We were introduced to the Dirac delta function $\delta(x)$. I have seen examples in applied courses where the system has potential $V(x)=c\delta(x)$ where $c>0$ is some constant. What does this actually mean physically? Also what is the point of scaling $\delta(0)=\infty$ by a constant?

Thank you. 

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up vote 4 down vote accepted

The only mathematical meaning I know is that $\delta$ is a measure (and not a function), such that, for every nicely behaved test function $\varphi$, $$ \int\varphi(x)\mathrm d\delta(x)=\varphi(0). $$ Physicists often write the LHS as $$ \int\varphi(x)\delta(x)\mathrm dx, $$ although the measure $\delta$ has no density with respect to Lebesgue measure, but this is the other formula they have in mind. Finally, if $\mu=c\delta$, one gets $$ \int\varphi(x)\mathrm d\mu(x)=c\varphi(0). $$

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What any mathematical object means "physically" depends entirely on how you are using mathematical objects to represent "physical" things.

For example, if $c \delta(x)$ it shows up as a term in energy density, then it contributes $c$ to the amount of energy contained in any region containing the point $x=0$.

Also what is the point of scaling $\delta(0) = \infty$ by a constant?

The equation $\delta(0) = \infty$ is* a fabrication -- it's a white lie meant to avoid concept of generalizing the notion of function.

(many -- myself included -- would argue the lie does more harm than good, but that's another topic)

The Dirac delta only has meaning inside an integral -- or in other situations related to or derived from this meaning, such as appearing in a (generalized) differential form -- and that meaning is $$ \int_{-\infty}^{+\infty} f(x) \delta(x) \, dx = f(0) $$ or some variation thereof, depending on technical concerns and convention. How $c \delta(x)$ differs from $\delta(x)$ is now obvious, by seeing how it affects the integral.

*: For the sake of honesty, I should point out that for some generalized notions of function, there is a corresponding generalized notion of evaluation in which the equation is true. But if you can understand the meaning of that, you wouldn't be confused by $c \delta(x)$.

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The constant is just there to change the value of the integral: $$\int c\delta(x)dx=c$$

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Scaling the Dirac function is scaling the integral which is 1 normally. It's also not really a function but a distribution.

Of course there is no such thing physically, but it's nonetheless useful. For example when measuring the echo of a room to replicate it electronically, one just has to fold a given signal with the impulse response, which is the echo of one delta spike. As there is no such thing, one has to try and create as sharp a spike as possible (an exploding balloon or just clapping your hands will often do).

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