# Rigorous definition of continuity on an open vs closed interval

1. Dec 1, 2011

### kahwawashay1

Let I be an open interval and f : I → ℝ is a function. How do you define "f is continuous on I" ?

would the following be sufficient? :

f is continuous on the open interval I=(a,b) if $\stackrel{lim}{x\rightarrow}c$ $\frac{f(x)-f(c)}{x-c}$ exists $\forall$ c$\in$ (a, b)

is this correct?

Also, what about the case of a closed interval I? In that case, can you just add to the above statement that:

$\stackrel{lim}{x\rightarrow}a^{+}$ f(x) = f(a)
and
$\stackrel{lim}{x\rightarrow}b^{-}$ f(x) = f(b)

???

2. Dec 1, 2011

### dextercioby

The definition you gave is for differentiability on an open interval. The definition for continuity is with delta and epsilon. The [ itex ] for lim looks like $\displaystyle{\lim_{x\rightarrow a}}$.

3. Dec 1, 2011

### kahwawashay1

But differentiability implies continuity?
Anyway, then would this be right:

f is continuous on the open interval I=(a,b) if |$\frac{f(x)-f(c)}{x-c}$ - f'(c)|< ε when |x-c|<δ $\forall$ c$\in$ (a, b)

and what about my use of the right and left hand limits for the case of a closed interval [a,b]? would that be correct?

4. Dec 1, 2011

### dextercioby

I was referring to the http://en.wikipedia.org/wiki/Continuous_function Weierstrass epsilon-delta definition for continuity.

5. Dec 1, 2011

### kahwawashay1

ohhhh nvm. it can be continuous on (a,b) but not differentiable, like the abs value of x.
Ok then it would just be:
f is continuous on (a,b) if $\displaystyle{\lim_{x\rightarrow c}}$ f(x) = f(c) $\forall$ c$\in$ (a, b)

right???

6. Dec 1, 2011

### dextercioby

Yes, but for that limit to make sense, you have to use the topology, that is compare values of f(x_1) and f(x_2) when x_1 and x_2 get arbitrarily close to each other. That's what the epsilon-delta definition does.

7. Dec 1, 2011

### kahwawashay1

we didnt learn about topology yet so idk what is f(x_1) and the like...