Thinking of a Continuous function

Question

While I was writing a proof of some theorem I need to think of a continuous function like this: $$F(x)=\begin{cases}\frac{1}{x(x+1)^{2}} & 0<x\leq 100\\ S & x=0 \end{cases}$$

Is is possible to put something in $S$ so $F$ will be continuous

Answer 1

HINT: What is $\lim_{x\to 0^+}\frac{1}{x(x+1)^2}$? What must $S$ be equal to in order to ensure continuity? What happens if $\lim_{x\to 0^+}\frac{1}{x(x+1)^2}$ is unbounded?

Answer 2

The answer is no.

Note that as x approaches $0$ from the right in $$F(x)=\begin{cases}\frac{1}{x(x+1)^{2}} & 0<x\leq 100\\ S & x=0 \end{cases}$$ $F(x)$ grows without bound, so no real number will make it continuous at $x=0$