Right continuous function with restricted domain

Question

If $f:\mathbb{R}\to \mathbb{R}$ is defined by $f(x):=0.5x$ for $x<1$, and $f(x)=x$ for $x\geq 1$, then is $f$ is clearly right continuous, but what if we restrict the domain to $[0,1]$, would the new function still be right continuous?

Answer 1

Well, are there any points $x_0\in[0,1]$ for which there is some $\epsilon>0$ such that for all $\delta>0,$ there is some $x\in[0,1]\cap(x_0,x_0+\delta)$ such that $$\left|f(x_0)-f(x)\right|\ge\epsilon?$$