Show that partial derivative is continuous

Question

Given a function $f \,:\, \mathbb{R}^2 \to \mathbb{R}$ I have shown the partial derivative $f_x$ exists and that $|f_x(x_1, y) - f_x(x_2, y)| \leq 16 |y|$.

Is it possible to show that $f_x$ is a continuous function using the epsilon-delta criterium?

Answer 1

No. Consider

$$ f(x,y)=\begin{cases} x,& \text{ if } y=0 \\ 0,& \text{ otherwise }\\ \end{cases} $$ That means $$ f_x(x,y)=\begin{cases} 1,& \text{ if } y=0 \\ 0,& \text{ otherwise }\\ \end{cases} $$

which is not continuous at any point $(x,0)$.