Joint Probability Distribution Function For some Joint Probability Function on two continuous random variables, $f(x,y)$, is it true that the integral of $f(t,t)dt$ from negative infinity to infinity is necessarily upper bounded by one? I thought that $4xy$ is a counter example, but I can't seem to visualize what I'm doing.
Assuming that $f(x,y)$ is the joint density of jointly continuous random variables $X$ and $Y$, it is straightforward to show that the density 
of $Z = Y-X$ is given by
$$g(z) = \int_{-\infty}^\infty f(x, z+x)\,\mathrm dx.$$
Thus your integral
$\displaystyle \int_{-\infty}^\infty f(t,t)\,\mathrm dt$ can be recognized
as $g(0)$. I will leave it up to you to decide whether it is
true that $g(0) \leq 1$ for all possible pairs of jointly continuous
random variables.
Note that if your variables are $X,Y$ then
$$
\int_\mathbb{R} f(t,t)dt = \mathbb{P}[X=Y=t] \leq 1
$$