I have a continuous function $f : \mathbb{R}^{n} \times \mathbb{R}^{m} \to \mathbb{R}^{k}$. If I assume that $f$ is $C^{1}$ with respect to its first argument such that $\nabla_{\! 1} f$ is continuous, I can write $$ f(x,y) = f(0,y) + \nabla_{\! 1} f(0,y)x + R(x,y), $$ where $R$ is a continuous remainder function.
Is the function $\lvert R(x,y) \rvert / \lvert x \rvert$ continuous?