Convexity in each argument and directional derivative

Question

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or infinite) $\lim\limits_{t\to 0+} \frac{f(t,t)-f(0,0)}{t}$?

Answer 1

Any convex function $f:\mathcal{D}\mapsto\mathbb{R}$ is directionally-differentiable (in any direction) on its relative interior, where $\mathcal{D}\subset\mathbb{R}^n$. Stated symbolically: $$\forall x \in \mathbf{relint}\mathcal{D}, v\in\mathbb{R}^n: f^\prime(x;v) \text{ exists}$$ and $|f^\prime(x;v)| < \infty$.

Just as a side note: If $f$ is convex, then the directional derivative $f^\prime(x;v)$ is convex in its second argument, the direction. For this, check out this other stack exchange question.