Continous functions questions

I'm not sure if my working is right for this one..

I have put:

For all epsilon > 0, there exists delta > 0, such that | f(x) - f(a) | < epsilon, | x - a | < delta

We can use a delta small enough to take epsilon = f(a)/2

| f(x) - f(a) | =< f(a)/2 = epsilon

Resulting in 3f(a)/2 > f(x) > f(a)/2

But I'm not sure if that's okay?

Next question:



Let x >= z, and let f(z) = inf{x : x E R}

Let f(z) = c

Take a small delta such that | x - z | < delta

Then I went to choose c such that |f(x) - c| < |f(x) - 1| < epsilon

Again, I feel like I'm not doing well enough here.

Basically OK, but needs sharpening up a bit.
Your definition of continuity is a bit sloppy.
We choose epsilon, f(a)/2, then there exists a delta such that if | x-a | < delta, then |f(x) - f(a) | < f(a)/2.
And continue.
Also, take care to use strict inequalities "<", or weak inequalities "<=", correctly. Sometimes it's not a significant part of a proof, but on other occasions the whole proof stands or falls on it.



What's z? and your set "inf{x : x E R}" doesn't make much sense.

Thanks for the help
Yeah I'm thinking that I should use weak inequalities for most of the first question, and then finally saying 3f(a)/2 >= f(x) >= f(a)/2, and then hence f(x) >= f(a)/2?

For the second part, I meant to say that I'm using the extreme value theorem, taking z in [0, infinity) such that all x >= z, hence f(z) = {f(x): x E R), I acidentally typed it up a bit wrongly

Well the question is asking for you to show f(x) > f(a)/2



Can you post your answer to the second question all together, as I'm not clear what's going on trying to combine that with what you said in your first post.

Also make sure you meet the criteria for the extreme value theorem in your usage of it.

Here is my answer for the second part:

Take z in [0, infinity)

Let x >= z, and let f(z) = inf{f(x) : x E R} I'm not sure if this should be x in R or x in [0, infinity)..

In essence yes, with the caveats I previously mentioned. You are asked to show f(x) > f(a)/2, not >= f(a)/2, the latter being a weaker condition.



OK, I choose z=5



But f(x) is defined already, in particular f(z) = f(5), so I can't define it again with that set.

Last post for today.

..

To my mind it's going to be very difficult to solve the 2nd question using just one idea (as it seems you're currently trying to do).

There are 2 separate things going on here - you've told f(x) is always > 0, and you're also told that f(x)->1 as x->infinity.

Now on any finite interval I, you can use the extreme value theorem to say f(x) attains it's bounds on I, and so since f(x) > 0, the bound on I is also > 0.

But your problem is that $[0, \infty)$ is NOT a finite interval.

So you're going to need to use what you're told about the behaviour of f(x) as x->infinity.

I would start by writiing down what it means to say that $f(x)\to 1$ as $x \to \infty$. Then use that to show that you can find an infinite interval $[M, \infty)$ on which we know f(x) > 1/2. Then extend to the full range $[0,\infty)$ by adding on a finite interval.

I think f(x) -> 1 as x -> infinity means f(x) is continous on all points between 0 and infinity, thus we can find c > 0

Taking epsilon = 1/2, and using an M in R, and letting x >= M, then I can have it such that | f(x) - 1 | =< 1/2 = epsilon, and from that we have f(x) >= 1/2

Now I can take the other part of the interval, [0.M], and find c in here, which would mean that f(x) >= 1/2 >= c > 0

I can find a z in the interval [0,M] such that f(z) = inf { f(m): m E [0,M] } > 0, and call this c

Then it would satisfy that f(x) >= 1/2 >= c as sup { f(m): m E [0,M] } would be =< 1/2?

No, this isn't what it means. There is no implication of continuity. A correct definition:

$f(x) \to a$ as if $\forall \epsilon > 0, \, \exists M \in \mathbb{R}$ such that $x > M \implies |f(x)-a| < \epsilon$.

This is basically correct, but it's very unclearly worded.

Not sure what you're trying to say here. Some of this isn't true, and what is true needs justification, which you haven't provided. I think the justification is supposed to be what's following, but you either need to put the justification first or rewrite the above to be more like "We still need to deal with the interval [0,M]. We shall show that we can find c > 0 such that ..."

If you're going to use a theorem, you should actually name it, and make it clear that any conditions are satisfied. e.g. "Since f is cts, and [0, M] is a closed interval, f attains its bounds on this interval. In particular, we can find z in [0, M] such that..."

There is no reason to assume c <= 1/2. E.g. f(x) = 1 everywhere (f is constant). You also seem to have switched from inf to sup but I assume that's a typo.

I would start by writiing down what it means to say that f(x)\to 1 as x \to \infty. Then use that to show that you can find an infinite interval [M, \infty) on which we know f(x) > 1/2. Then extend to the full range [0,\infty) by adding on a finite interval.