# Continuity of Lipschitz constant over multi-variable function

Let $$f\in C^0(I\times \mathbb{R})$$,$$I=[\xi,\xi+a]$$,$$a>0$$, and suppose that $$\max_{x\in I}|f(x,\eta)|=M<\infty$$ for some given $$\eta\in\mathbb{R}$$. Also, suppose $$f$$ is differentiable with respect to $$y$$ with $$|f_y(x,y)|\leq K|f(x,y)|$$ uniformly on $$x\in I$$. Then for any given $$x\in I$$, $$f(x,\cdot)\in C^1(\mathbb{R})$$ is locally Lipshitz, i.e. $$\exists \delta_x>0 \exists L_x\geq 0,(y_1,y_2\in[\eta-\epsilon_x,\eta+\epsilon_x]\rightarrow\\\forall(y_1,y_2)\in\mathbb{R}^2(|f(x,y_1)-f(x,y_2)|\leq L_x|y_1-y_2|)).$$

The question is ,

Is such $$L_x$$ continuous with respect to $$x$$?

and I suppose such $$L_x$$ is taken as a infimum in kind that are possible.

• The function $f$ given by $f(x,y)=e^y$ satisfies all the conditions (say with $\xi=0,a=1,\eta=0,M=1,K=1$), but is not Lipschitz in $y$. – Iosif Pinelis Jan 24 at 1:34
• @losif Yes, you are right, I changed my argument into locally Lipschitz. – An Jin Jan 24 at 2:04

E.g., let $$\xi:=-1$$ and $$a:=2$$, so that $$I=[-1,1]$$. Let $$f(x):=1+y(1-|y|/x)^2\,1_{x>0,\,|y| for $$x\in I$$ and real $$y$$. Let $$\eta:=0$$, so that $$M=0$$. Let $$K=1$$. Then all your conditions on $$f$$ hold.
Yet, $$L_x=1_{x>0}$$ is not continuous in $$x\in I$$.
Here is the graph $$\{(x,y,f(x,y))\colon -0.2\le x\le1,\,-1.2\le y\le1.2\}$$: