Let $d_F(x,y)$ be the Bregman divergence of $x$ and $y$. By the definition, Bregman divergence requires strictly convex function $F$. Let $x=[x_1,x_2]^\top$ be a $2$-dimensional vector, I was wondering if $F(x)=\lim_{\epsilon\rightarrow 0} x_1^2+\epsilon x_2^2, \ \epsilon>0$, then, is $d_F(x,y)$ still a valid Bregman divergence?
In other words, is $F(x)=\lim_{\epsilon\rightarrow 0} x_1^2+\epsilon x_2^2$ is a strictly convex function?
A simpler question is is $f(x)=\lim_{\epsilon\rightarrow 0}\epsilon x^2$ a strictly convex function?