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In real analysis courses, students are often taught a theorem which states that:

If $f$ is a real valued continuous function on $[0,1]$, then $f$ is bounded there

and the example $f(x)=\frac{1}{x}$ is often used to illustrate this point. I've seen the proof using compactness, but the theorem itself never made much sense to be because isn't $f(x)=\frac{1}{x}$ discontinuous at $0$ since $$\lim_{x\rightarrow 0+}f(x) \neq \lim_{x\rightarrow 0-}f(x)$$.

The theorem is said not to hold if the interval $(0,1)$ was used instead but it seems really counterintuitive to me.

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1 Answer 1

The function $f(x)=\frac{1}{x}$ is continuous on the interval $(0,1)$ but not bounded so it is a counterexample. It is also true that the theorem fails for intervals of the form $(0,1]$ and $[0,1)$ for a similar reason. To see this consider functions like $\frac{1}{x}$ and $\frac{1}{1-x}$ respectively.

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I never understood why it was bounded on $[0,1]$ since the discontinuity is at $0$ –  Millardo Peacecraft Aug 6 at 22:35
    
It is continuous where defined on $[0,1]$, but is undefined at $0$. However, it is not bounded on $[0,1]$, and so is a rather poor choice of example (Being that it isn't an example at all). –  qaphla Aug 6 at 22:37
    
It is not bounded on $[0,1]$ because it tends to $\infty$ as $x$ tends to $0$. You could redefine the function at $x=0$ but it could never be continuous. The theorem says the function must be continuously defined on all of $[0,1]$ to hold. –  user71352 Aug 6 at 22:37
    
@user71352 I'm sorry I meant to say continuous and not bounded. As I pointed out in the question the left and right hand limits don't agree so it shouldn't be continuous there or am I missing something? –  Millardo Peacecraft Aug 6 at 22:40
    
@MillardoPeacecraft Well if you are defining the function only on $[0,1]$ then there really is no left limit at $x=0$. However, when left and right limits do not agree this means the function is not continuous at that point. The function is an example of being continuous and not bounded on $(0,1)$ since the function is defined and continuous in this region but not bounded. –  user71352 Aug 6 at 22:56

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