Epsilon-delta Watch

coconutseasalt
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I have a problem that I really don't know how to begin to solve. I am assuming that I would need to construct an epilson-delta proof of some sort but I am really stuck as to where I should begin. Any nudges in the right direction would be much appreciated, so thanks in advance to anyone who helps

Let f : [a, b] → R be a continuous function and assume that f(c) < 0 for some c ∈ [a, b].
Show that there exists a δ > 0 such that 3f(c) < f(x) < −f(c) holds for all x ∈ [a, b]
which satisfy |x − c| < δ
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LightBlueSoldier
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(Original post by coconutseasalt)
I have a problem that I really don't know how to begin to solve. I am assuming that I would need to construct an epilson-delta proof of some sort but I am really stuck as to where I should begin. Any nudges in the right direction would be much appreciated, so thanks in advance to anyone who helps

Let f : [a, b] → R be a continuous function and assume that f(c) < 0 for some c ∈ [a, b].
Show that there exists a δ > 0 such that 3f(c) < f(x) < −f(c) holds for all x ∈ [a, b]
which satisfy |x − c| < δ
Subtract f(c) from the inequality
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ghostwalker
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(Original post by coconutseasalt)
...
I'd start by stating the \epsilon,\delta definition of continuity for when x=c.
Change the modulus signs for the range into two inequalities.
Make a suitable choice for \epsilon (that's the key) and your result drops out almost immediately.
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coconutseasalt
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(Original post by ghostwalker)
I'd start by stating the \epsilon,\delta definition of continuity for when x=c.
Change the modulus signs for the range into two inequalities.
Make a suitable choice for \epsilon (that's the key) and your result drops out almost immediately.
Thank you very much for your help! I think I've got the answer now
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