# Thread: Continuity definiton problem

1. ## Continuity definiton problem

Show that $f: A \rightarrow R^{m}$ is continuous at $x_0$ iff for every $\epsilon > 0$, there exist $\delta > 0$ such that $|| x-x_0 || \leq \delta$, we have $||f(x)-f(x_0)|| \leq \epsilon$

So I can replace the definition < with $\leq$?

2. Originally Posted by tttcomrader
Show that $f: A \rightarrow R^{m}$ is continuous at $x_0$ iff for every $\epsilon > 0$, there exist $\delta > 0$ such that $|| x-x_0 || \leq \delta$, we have $||f(x)-f(x_0)|| \leq \epsilon$
So I can replace the definition < with $\leq$?
I am not sure that I follow your question.
Do note that the tradition definition is for “<”.
However, it is also $\left( {\forall \varepsilon > 0} \right)\left[ {\frac{\varepsilon }
{2} < \varepsilon } \right]$
, that is for all.
So $\left\| {f(x) - f(x_0 )} \right\| \leqslant \frac{\varepsilon }
{2} \Rightarrow \quad \left\| {f(x) - f(x_0 )} \right\| < \varepsilon$

3. I asked the professor today and he said this problem is suppose to show that the two definitions for continuity are equiv.

Well, from the last post I understand how to go from $\leq$ to <

How to do this the other way around is killing me...

Any hints? Thanks.