The problem is to prove the following:
Suppose f(x) is continuous on the
closed interval
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) xThis statement has a delta-epsilon contract that goes along with it. Before continuing reading this, try to write out that contract for yourself.> c
The contract goes like this: For every
|f(x) - f(c)| £ ewhenever
|x - c| £ dIn 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
Step 2: Apply the contract. We set e to something
less than f(c) but still positive.
Whenever
Remember the condition given in the problem? It is that for some
d
it should be true that f(x) is positive whenever
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
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