# Would you use Trichotomy?

1. Nov 3, 2008

### tronter

Let $$X$$ be a metric space, let $$a \in X$$ be a limit point of $$X$$, and let $$f: X \to \mathbb{R}$$ be a function. Assume that the limit of $$f$$ exists at $$a$$. Fix $$t \in \mathbb{R}$$. Suppose there exists $$r > 0$$ such that $$f(x) \geq t$$ for every $$x \in B_{r}(a) \backslash \{a \}$$; then $$\lim_{x \to a} f(x) \geq t$$.

How would you prove this? Would you use Trichotomy?

2. Nov 3, 2008

### Office_Shredder

Staff Emeritus
Re: Continuity

Find a sequence xk such that xk converges to a. Then what can you say about f(xk) for each k?

3. Nov 3, 2008

### boombaby

Re: Continuity

you will get a contradiction pretty easy if lim f(x)<t.