I am working on the function $f:[0,1) \to [0,1]$ defined by $f(x)=\dfrac{x}{x+1}$ for each $x\in [0,1)$.
It is clear that $f$ is well defined and continuous.
Now consider another function given by $ F(x)= \begin{cases} f(x), &x\in[0,1), \\ 1, & x = 1. \end{cases} $ It is clear that $F$ is an extension of $f$ into $[0,1]$. But it is not clear to me that $F$ is not continuous at $x=1$, even though it is defined at $x=1$. Please I need more clarification about this.