Limsup of a continuous function

Question

I'm trying to familiarize with $\limsup$, but I can not understand this example.

Consider the function $f:[0,1]\rightarrow \mathbb{R}$, $f(x)=x+1$. Clearly $f$ is continuous and $f(0)=1$.

Is it true that $\limsup\limits_{x\rightarrow 0^+}f(x)$=1?

My answer would be "yes": $\limsup\limits_{x\rightarrow 0^+}f(x)=\lim\limits_{\epsilon\rightarrow 0}(\sup\{f(x):x\in(0,\epsilon)\})=\lim\limits_{\epsilon \rightarrow 0}f(\epsilon)=1$

Is it right?

Answer 1

Yes, you are right. Since $f$ is continuous at $0$ we have

$\limsup\limits_{x\rightarrow 0^+}f(x)=\lim\limits_{x\rightarrow 0^+}f(x)=f(0)=1$.