Is there any redundancy here?

Question

Let $\Phi$ be the class of all functions $\phi:[1, \infty) \rightarrow[0, \infty)$ satisfying the following properties:

1. $\phi$ is lower semi-continuous;
2. $\phi(1)=0$;
3. for each sequence $\left\{t_{n}\right\} \subseteq(1, \infty), \lim _{n \rightarrow \infty} \phi\left(t_{n}\right)=0$ if and only if $\lim _{n \rightarrow \infty} t_{n}=1$.

Is there any redundancy between those conditions? I mean, doesn't $1$ and $3$ implies $2$?

Answer 1

Yes. Indeed by (3) there exists $\lim_{x\to 1}\phi(x)$, hence $$ 0 \overset{(hp)}{\le} \phi(1) \overset{(1)}{\le} \liminf_{x\to 1}\phi(x)\overset{(above)}{=}\lim_{x\to 1}\phi(x) \overset{(3)}{=}0. $$