Induced continuity or differentiablity on a line on a square

Question

Let $k\in\mathbb N_0$, $A:=(-1,1)$ and $f:A^2 \to \mathbb R$ be a function with the following properties:

1. For every fixed fixed $y \in A$ the function $f(\cdot,y)$ is $C^\infty$.
2. For every fixed fixed $x \in A\setminus\{0\}$ the function $f(x,\cdot)$ is $C^k$.

Question: Is $f(0,\cdot)$ as well $C^k$?

Answer 1

This is false when we consider

$$f(x,y) = \sqrt{x^2+4y^2} - \sqrt{x^2+y^2}$$

and $k = 1$.

This is also false for $k = 0$ when we consider $$g(x,y) := \frac{4y}{\sqrt{x^2+4y^2}}- \frac{y}{\sqrt{x^2+y^2}}$$

which is defined so that $g(0,0) = 0$, note that $g(0,y) = \frac{y}{|y|}$ when $y$ is not $0$; thus this function is your counterexample. Thus the above question is false for all $k \geq 0$.