Box 3.0d: Solution to Continuity Proof Problem
© 1997 by Karl Hahn
The problem is to prove the following: Suppose f(x) is continuous on the closed interval a ≤ x ≤ b, and suppose that there is a point, c, on that interval for which f(c) > 0. Prove that there exists a δ such that f(x) > 0 for all x in the interval, c - δ < x < c + δ.
Step 1: Apply the definition of continuity. The function, f(x), is continuous on the entire interval and the point, c is on that interval, so f(x) is continuous at c. What does this mean according to the definition of continuity? It means that f(c) exists and
f(c) = lim f(x) x → cThis statement has a delta-epsilon contract that goes along with it. Before continuing reading this, try to write out that contract for yourself.
The contract goes like this: For every ε > 0, there is a δ > 0 such that
|f(x) - f(c)| ≤ εwhenever
|x - c| ≤ δIn other words, if you tell me how close you need f(x) to be to f(c), I can tell you how close to make x to c in order to make it so.
We know that f(c) > 0 because that's given in the problem. That means that it falls on the positive side of zero. The problem asks us to show that if x is within some δ of c, no f(x) falls on the other side of zero. You already have the hint that you should try setting ε to something less than f(c). Before continuing reading this, see if you can't complete the problem with the information you have so far.
Step 2: Apply the contract. We set ε to something less than f(c) but still positive. Whenever f(x) ≤ 0, we know that |f(x) - f(c)| ≥ f(c). Can you see why? If not, try plugging in a positive number for f(c) and a negative number for f(x) and see what happens to the above inequality. And that means that if |f(x) - f(c)| ≤ ε, then f(x) cannot be negative or zero. So it must be positive.
Remember the condition given in the problem? It is that for some δ it should be true that f(x) is positive whenever c - δ < x < c + δ. Isn't that the same as saying |x - c| < δ? Isn't that the same wording as our contract? And doesn't the contract stipulate that if |f(x) - f(c)| ≤ ε then such a δ > 0 must exist that always makes that happen? And we know that when it happens, f(x) must be positive. That completes the proof.
If you are asked to do a proof like this on an exam, you don't have to be as wordy as I have been here. The main points are: 1) that if you choose ε < f(c), that implies that a nonpositive f(x) fails the ε test, and 2) that the epsilon-delta contract implied by continuity guarantees that a δ exists that causes every x in the δ-interval to meet the ε test.
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