2 dimensional random variable uniform over 1 dimensional region.

Question

I am struggling with following problem: Given random vector $(X, Y)$ with uniform distribution over diagonal of unit square which connects points $(0, 0)$ and $(0,1)$, find CDF and marginal distributions. I solved some similar problems by finding density function, and then calculating adequate integrals. Since the diagonal is $\{(a, a) : a \in [0, 1]\}$, my initial thought was that densitiy is just $$f(x, y) = \mathbb{1}_{[0, 1]}(x)*\mathbb{1}_{\{x\}}(y)$$ Integrating it over $\mathbb{R}^2$ yields $0$ obviously. I tried calculating CDF directly. $$F(x, y) = \mathbb{P}[X\leq x, Y \leq y] = \cases{\min(x, y, 1) \text{ if } x, y \geq 0 \\ 0 \text{ otherwise} }$$ F is continous, $f(x, y) = \frac{\partial^2 F}{\partial x \partial y} \equiv 0$, seems to be wrong too. So now I am confused - how can 2 dimensional random variable have uniform distribution over 1d region, and what am I doing wrong?