Continuous $(x,y) \to k(x,y)$ with "discontinous slope-behavior" at one $x\to k(x,y)$-slice: Is this possible?

Question

My question is if it is possible to construct a continuous function $k\colon[a,b]\times [c,d] \to \mathbb{R}$ such that for each $v\in (c,d]$ the function $k(\cdot,v)$ its slope (i.e. derivative) is everywhere defined and finite but for $v=c$ its slope is also everywhere defined but at some point infinite ?

Somehow this seems surprisingly nontrivial to disprove, unless I overlooked something obvious.

Answer 1

The function $k:[0,1]\times[0,1]\rightarrow\mathbb{R}$, $k(x,y)=\sqrt{x+y}$ satisfies your conditions.