# Analysis: Continuous Functions

• Shackleford
In summary, the conversation discusses various mathematical concepts and clarifications related to defining sequences, continuity, and function values. Some errors in notation and approaches are pointed out and corrected, and the importance of defining a proper domain is emphasized.
Shackleford
I did the work. I'm not sure on some of these.

I think for (c) I need to make D = (0, infinity)

http://i111.photobucket.com/albums/n149/camarolt4z28/1-3.png

http://i111.photobucket.com/albums/n149/camarolt4z28/2-3.png

http://i111.photobucket.com/albums/n149/camarolt4z28/3-1.png

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Some remarks. You're defining sequences wrong. You're saying (for example):

$$x_n=x$$

You can't do this. Your sequence must be dependent on n and not on x. So you need to write

$$x_n=n$$

or

$$x_n=\frac{1}{n}$$

Another remark. In (h), the function 1/f isn't well-defined. Indeed, it can happen that we divide by 0! So you can't use that argument there (or at least not directly). Does it change anything if f(x)=0 for an x??

For (c). You can't really do that. Indeed, $f(+\infty)$ isn't well-defined. You need c in the reals!

Yeah, I actually thought about that earlier. I knew I was being sloppy in my notation. Let me change that.

Oh, in (c), I made an error. It should be pi/4. Is my approach correct?

In (h), I said f(c) does not equal 0. I'm not really sure how to do this problem.

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Shackleford said:
Yeah, I actually thought about that earlier. I knew I was being sloppy in my notation. Let me change that.

Oh, in (c), I made an error. It should be pi/4. Is my approach correct?

The idea might be correct. But the use of infinite must be evaded.

In (h), I said fc does not equal 0. I'm not really sure how to do this problem.

That's the point, you can't assume that f(c) doesn't equal zero! For all we know, f can be the zero function! (hint hint)

micromass said:
The idea might be correct. But the use of infinite must be evaded.
That's the point, you can't assume that f(c) doesn't equal zero! For all we know, f can be the zero function! (hint hint)

Why must the use of infinite be evaded? One of the definitions in a theorem for continuity says,

If (xn) is any sequence in D such that (xn) converges to c, then limit of f(xn) as n tends to infinity is f(c). Isn't my example a valid counterexample?

Shackleford said:
Why must the use of infinite be evaded? One of the definitions in a theorem for continuity says,

If (xn) is any sequence in D such that (xn) converges to c, then limit of f(xn) as n tends to infinity is f(c). Isn't my example a valid counterexample?

Your theorem applies if c is a real number. You can't choose c=infinity. Furthermore, f(infinity) is not well-defined.

micromass said:
Your theorem applies if c is a real number. You can't choose c=infinity. Furthermore, f(infinity) is not well-defined.

You're talking about (c)? What about (f)?

For (c), it's not f(infinity). The argument is not infinity. Are you talking about (f)?

In (c), what is your c?? What is f(c)??

micromass said:
In (c), what is your c?? What is f(c)??

Haha. In (c), c = pi/4. f(c) is root2/2.

I picked an f(c) value with a sequence that did not converge to that c. That's what the theorem states. If you know that f is continuous, then if a sequence converges to some c value, then the limit of f(of that sequence) will equal f(c).

The statement is not necessarily true because it starts with the function value. It's backwards with respect to the theorem.

Shackleford said:
Haha. In (c), c = pi/4. f(c) is root2/2.

I picked an f(c) value with a sequence that did not converge to that c. That's what the theorem states. If you know that f is continuous, then if a sequence converges to some c value, then the limit of f(of that sequence) will equal f(c).

The statement is not necessarily true because it starts with the function value. It's backwards with respect to the theorem.

Uuuh, that's not what I understand when I read (c). You better rephrase that answer...

micromass said:
Uuuh, that's not what I understand when I read (c). You better rephrase that answer...

Really? Is what I just wrote correct? I thought I construed it properly mathematically on the paper.

For (h), I suppose it's important to define a proper domain. If f is continuous at some point c and f(c) = 0 , then g(x) doesn't necessarily have to be continuous. But that's just a wild guess.

## What is a continuous function?

A continuous function is a type of mathematical function where the output values vary smoothly as the input values change. This means that there are no sudden jumps or breaks in the graph of the function, and it can be drawn without lifting your pencil from the paper.

## How do you determine if a function is continuous?

To determine if a function is continuous, you can use the three-part definition of continuity: 1) the function is defined at the point in question, 2) the limit of the function at that point exists, and 3) the limit is equal to the value of the function at that point. If all three parts are true, then the function is continuous at that point.

## What are some examples of continuous functions?

Some common examples of continuous functions include linear functions, quadratic functions, trigonometric functions, exponential functions, and logarithmic functions. These functions can be represented by smooth, unbroken curves on a graph.

## What is the importance of continuous functions in mathematics?

Continuous functions are important in mathematics because they allow us to model and analyze real-world situations with precision and accuracy. They also provide a foundation for other key concepts in calculus and analysis, such as limits, derivatives, and integrals.

## Can a function be continuous at some points and discontinuous at others?

Yes, a function can be continuous at some points and discontinuous at others. This is known as a piecewise continuous function, where different parts of the function behave differently. For example, a step function is continuous at each step, but not continuous as a whole.

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