(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!

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# Homework Help: Continuity implies boundedness in an interval proof

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