# Real Analysis: Function Continuity

Let $$A, B \subset \mathbb{R}$$, let $$f : A \rightarrow B$$, and let $$x_0 \in A$$. Then $$f$$ is continuous at $$x_0$$ if and only if $$\lim_{x \rightarrow x_0} f(x) = f(x_0)$$.

I proved the first part

$$f$$ is continuous at $$x_o$$ if $$\lim_{x \rightarrow x_0} f(x) = f(x_0)$$

using "$$f$$ is continuous at $$x_o$$ in $$A$$ ($$dom(f)$$) if, for every sequence {$${x_n}$$} in $$A$$ ($$dom(f)$$) converging to $$x_0$$, we have $$\lim_n f(x_n) = f(x_0)$$"

Moreover, I tried proving the second part, which is

$$\lim_{x \rightarrow x_0} f(x) = f(x_0)$$ if $$f$$ is continuous at $$x_o$$

using the other definition of continuity that involves $$\delta > 0$$ and $$\epsilon >0$$, which eventually proves that $$\lim_n f(x_n) = f(x_0)$$.

My question is, is it correct proving the second part of the statement using aforementioned definition? or are there any other ways to prove the second part?

To prove the second part, suppose that it is not true. Then for some sequence $$\{x_n\}$$ converging to $$x_0$$, it is not true that $$f(x)$$ converges to $$f(x_0)$$, since otherwise for every sequence $$\{x_n\}$$ converging to $$x_0$$, $$f(x)$$ converges to $$f(x_0)$$, then $$\lim _{x \to x_0} f(x)=f(x_0)$$. Therefore $$f$$ is not continuous at $$x_0$$, which is a contradiction.

There is nothing to prove since it is just a definition that $$f(x)$$ is continuous at $$x=x_0 \iff \lim_{x\to x_0} f(x)=f(x_0)$$.

What we need to prove, in general, is that an assigned function $$f(x)$$ is continuous at a point according to the given definition.

Refer for example to the related

• I think that this answer misses the point. There are numerous definitions of continuity that one encounters in real analysis, e.g. via $\lim_{x \to x_0} f(x) = f(x_0)$, having $\lim_{n \to \infty} f(x_n) = f(x_0)$ for all sequences $x_n \to x$, or that $f^{(-1)}(\mathcal{O})$ is open for open sets $\mathcal{O}$. The fact that the asker is trying to prove the equivalence of the first two definitions is in the question. (-1)
– user296602
Commented Nov 14, 2018 at 3:00
• @TBongers In that specific case the initial statement to prove seems just the first definition you also refer to. Maybe you are right, but then the question should be formulated in a more clear way about that. Let help the asker with that.
– user
Commented Nov 14, 2018 at 3:26
• Yes, sorry for the vague statements. What I'm trying to ask is that whether or not there are other ways to prove the initial statements other than using the definitions of continuity themselves. Hope this clarifies. Commented Nov 14, 2018 at 3:30
• @JeffrySantosa My point is that the initial statement is simply the (one) definition of continuity and we don’t need to prove it. If you need to prove the equivalence between two or more different definitions, as claimed here by TBonger, then you should clearly state that.
– user
Commented Nov 14, 2018 at 3:36