Let $a,b\in \mathbb R^+$ and let $f:A\times\mathbb R$ such that $$f(x,y)=\frac{y}{\cos\left(a x\right)+\frac{a}{\sqrt{a^2+b^2}+b}\sin\left(a x\right)}$$ where $A=\{x|\cos\left(a x\right)+\frac{a}{\sqrt{a^2+b^2}+b}\sin\left(a x\right)\neq0\}$. My question: is this a locally lipschitz function?
I think that this is true, because the denominator of $f$ is continuous in $A$ and $$\left|f(x_1,y_1)-f(x_2,y_2)\right|\leq \sup_{x\in A}\left\{\frac{1}{\cos\left(a x\right)+\frac{a}{\sqrt{a^2+b^2}+b}\sin\left(a x\right)}\right\} \cdot \left|y_1-y_2\right|$$ Is it true?