(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If a function [itex]f[/itex] is continuous at a point [itex]x[/itex], then [itex]f[/itex] is bounded on some interval centered at [itex]x[/itex]. That is, [itex]\exists M \geq 0[/itex] s.t. [itex]\forall y[/itex], if [itex]|x - y| < \delta[/itex], then [itex]|f(y)| \leq M[/itex]

2. Relevant equations

3. The attempt at a solution

Let [itex]\varepsilon > 0[/itex]. Since [itex]f[/itex] is continuous at [itex]x[/itex], [itex]\exists \delta > 0[/itex] s.t. [itex]\forall y[/itex], if [itex]|x - y| < \delta[/itex], then [itex]|f(x) - f(y)| < \varepsilon[/itex]. Now,

[itex]|f(x) - f(y)| < \varepsilon \iff[/itex]

[itex]- \varepsilon < f(x) - f(y) < \varepsilon \iff[/itex]

[itex]f(x) - \varepsilon < f(y) < f(x) + \varepsilon[/itex].

Stated differently,

[itex]f(y) \in (f(x) - \varepsilon, f(x) + \varepsilon)[/itex].

We are trying to find an [itex]M \geq 0[/itex] s.t. [itex]|f(y)| < M[/itex], or [itex]f(y) \in (-M, M)[/itex]. This is where I get a little stuck. I realize that we may choose any [itex]\mu > 0[/itex] and say that [itex]M = f(x) + \varepsilon + \mu[/itex]. But I run into trouble when doing this. I want something like [itex]M = f(x) \pm (\varepsilon + \mu)[/itex]. This would cover the interval that I am trying to get [itex]M[/itex] into, but I don't exactly know how to say it (except by the way just mentioned of course). Is there a way to write this as a SINGLE value (as opposed to the [itex]\pm[/itex] showing up)?

EDIT: What about [itex]M = f(x) + |\varepsilon + \mu|[/itex]?

Thank you for your help!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Continuity implies boundedness in an interval proof

**Physics Forums | Science Articles, Homework Help, Discussion**