Proving Property of a Continuous Function

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Homework Statement

Homework Equations

Continuity @ v₀

The Attempt at a Solution

Using the epsilon delta definition of continuity:

If we choose epsilon such that epsilon < a, then |f(v) - f(v0)| < a.
So f(v) is in the interval (f(v0) - a, f(v0) + a).
Only half of this interval is what I want though.
I may be doing the whole thing wrong… I would really appreciate any help

edit: Actually, I'm pretty sure my reasoning is completely wrong, so I'll think about it some more...

 

[lanedance]

i think you're on the right track, so just clarifying what you've got:

you know f(v0)>a, so define M by f(v0)-a=M>0

now use the continuity of f and choose e>0, such that

|f(v0)-f(v)|<e<M

then there exist d>0 such that for |v0-v|<d, then |f(v0)-f(v)|<M/2

if f(v) > f(v0) we're fine, if f(v)< f(v0) then

|f(v0)-f(v)| = f(v0)-f(v) < M = f(v0) -a

then
f(v) > a

(note drawing a picture really helps for this one)