Question

In: Advanced Math

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3...

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous.

b) theorem 17.3 states that if f is continuous at x0, then |f| is continuous at x0. is the converse true? if so, prove it. if not find a counterexample.

hint: use counterexample

Solutions

Expert Solution

a) Let f(x) be continuous at any point c.

Then there exists a sequence, converging to , such that the sequence converges to .

Then by definition, for , such that for

for and ((By sequential criteria for continuity.))

Now by triangle inequality we know that :

for

Hence for a sequence converging to , the sequence converges to .

Therefore, by sequential criteria for continuity, is continuous.

b) Consider the function

It can easily be proven that this function is not continuous at any point.

But at every point. And is hence, continuous at each and every point.

Therefore, the converse may not be true. That is, if is continuous at a point , that does not mean that will be continuous at .

Colby Messinger answered 1 year ago

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