Simple calcul of a limit superior

Question

What is the limit superior of : $$ \limsup_{x\rightarrow 10} 1*\mathbb{I}_{x<10}(x)+2*\mathbb{I}_{x=10}(x)+\frac{1}{2}*\mathbb{I}_{x>10}(x) ? $$ If I take the definition of a limit sup I arrive to 2, but I have a little doubt on my procedure.

Answer 1

Note $f(x)=1*\mathbb{1}_{x<10}(x)+2*\mathbb{1}_{x=10}(x)+\frac{1}{2}*\mathbb{1}_{x>10}(x)$. You have

$$f(x)=\begin{cases} 1 * 1 + 2 * 0 + \frac{1}{2} * 0 &= 1 &\text{for } x<10\\ 1 * 0 + 2 * 0 + \frac{1}{2} * 1&= \frac{1}{2} &\text{for }x>10 \end{cases}$$ Hence $$\limsup\limits_{x \to 10} f(x)= 1$$. As for the computation of a limit, you don't take into account the value of the function at the point itself.