# Bounded function implies limit is bounded

• miren324
In summary, the problem is to prove that if a function f has a limit L at a point c which is an accumulation point of its domain D, and f(x) is bounded by a and b for all x in D except for c, then a<=L<=b. The speaker has tried to prove by contradiction and using the analogy of sequences and functions, but is unsure of how to construct a formal proof. Their intuition is that the limit of f as x approaches c is "near" f(c), but they are not sure how to translate this into a general proof.
miren324
Here's the problem:
Let f:D-R and c in R be and a accumulation point of D, which is a subset of R. Suppose that a<=f(x)<=b for all x in D, x not equal to c, and suppose that limx$$\rightarrow$$c f(x) = L. Prove that a<=L<=b.

I'm having trouble here. I've tried to prove by contradiction, by assuming that L<a, then after a contradiciton, assuming L>b. This led through a lot of junk and I ended up back where I started. I am unsure how to construct a proof using the analogous relationship of sequences and functions.

Any help would be greatly appreciated.

What is your intuition? If you were going to explain informally the reason, what would you say?

It just makes sense that it would be true. I know that if x does not equal c, the limit as x approaches c of f(x) is L, and L is "near" f(c-.00001) and f(c+.00001) and f(c-.000000001), etc. But I only know that from experience with linear equations. I'm not sure how to translate this into a formal proof for all functions, domains, and limits.

## 1. What does it mean for a function to be bounded?

A bounded function is one where the values of the function are limited or confined within a certain range or interval. This means that the function does not have values that go to infinity or negative infinity.

## 2. How does a bounded function imply that the limit is also bounded?

If a function is bounded, this means that the values of the function are limited. Since the limit of a function is the value that the function approaches as the input approaches a certain value, if the function is bounded, the values will not exceed a certain range as the input approaches that value. Therefore, the limit will also be bounded.

## 3. Can a function be bounded but not have a bounded limit?

Yes, a function can be bounded but not have a bounded limit. This can happen if the function has oscillations or fluctuations that prevent the values from approaching a specific value as the input approaches a certain value. In this case, the function is still bounded, but the limit is not bounded.

## 4. How does the boundedness of a function affect its continuity?

The boundedness of a function does not necessarily affect its continuity. A function can be both bounded and continuous, or it can be bounded but not continuous. Continuity is determined by the behavior of a function at a specific point, while boundedness is determined by the overall behavior of a function within a certain interval.

## 5. Are there any real-life applications of bounded functions and their limits?

Yes, there are many real-life applications of bounded functions and their limits. For example, in finance and economics, functions that model the growth or decline of investments or markets are often bounded as they cannot have infinite growth or decline. In physics and engineering, functions that describe physical processes or systems are often bounded as they cannot have infinite values.

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