- #1
gimpy
- 28
- 0
Im having a little trouble with this question.
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 I am 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?
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 I am 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?