Im having a little trouble with this question.(adsbygoogle = window.adsbygoogle || []).push({});

If f is continuous at [tex]c[/tex] and [tex]f(c) < 5[/tex], prove that there exists a [tex]\delta > 0[/tex] such that [tex]f(x) < 7[/tex] for all [tex]x \in (c - \delta , c + \delta)[/tex]

So we are given that f is continuous at c.

So [tex]\lim_{x \to c}f(x) = f(c) < 5[/tex]

[tex]\forall \epsilon > 0 \exists \delta > 0[/tex] such that whenever [tex]|x - c| < \delta[/tex] then [tex]|f(x) - f(c)| < \epsilon[/tex]

[tex]|x - c| < \delta[/tex]

[tex]-\delta < x - c < \delta[/tex]

[tex]c - \delta < x < c + \delta[/tex]

Ok now im getting lost..

I know i have to do something with [tex]|f(x) - f(c)| < \epsilon[/tex]

maybe

[tex]|f(x) - 5| < \epsilon[/tex]

[tex]-\epsilon < f(x) - 5 < \epsilon[/tex]

[tex]5 - \epsilon < f(x) < 5 + \epsilon[/tex]

we want [tex]f(x) < 7[/tex] so .... am i on the right track??? How can i find the [tex]\delta > 0[/tex] that satisfies this?

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

Dismiss Notice

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!

# Continuous function.

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