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?

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# Continuous function.

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