Convergence and continuity question

In summary, the proof concludes that f_n(x_n) converges to f(x) if and only if f is continuous at all the values on (x_n) and x is an element of D.
  • #1
buzzmath
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0
Can anyone help with this question?

Let (f_n) be a sequence of continuous functions on D a subset of R^p to R^q s.t. (f_n) converges uniformly to f on D, and let (x_n) be a sequence of elements in D which converges to x in D. Does it follow that (f_n(x_n)) converges to f(x)?

My proof goes like this:
Since (f_n) are all continuous and uniformly converge on D to f we know that f is continuous on D. By a thm. So f is continuous at all the values on (x_n) and is continuous at x. Since (f_n) is uniformly convergent to f there exist a natural number k(e) s.t. for all e>0 n>=k(e) x an element of D that ||f_n(x)-f(x)|| < e but since (x_k) and x are both elements of D we know ||f_n(x_k)-f(x_k)||<e for all natural numbers k. So we know f_n(x_k) converges to f(x_k) (By a thm that says if f is continuous at a then if (x_n) is any sequences in D(f) that converges to a,k the the sequence (f(x_n)) converges to f(a)) We know that (x_n) an element of D is a sequence which converges to x Thus f(x_n) converges to f(x). Thus f_n(x_n) converges to f(x_n) which converges to f(x). Thus (f_n(x_n)) converges to f(x).

My problem is I can see that if (f_n(x)) converges to f(x) and if f(x_n) converges to f(x) then the result seems clear that (f_n(x_n)) converges to f(x) but in my proof it doesn't seem clear to just say that. Does this make sense or is there a more precise way to say it?

thanks
 
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  • #2
If you want to be rigorous you should use the epsilon delta definition. You can get |f(x)-f_n(x_n)|=|f(x)-f(x_n)+f(x_n)-f_n(x_n)| <=|f(x)-f(x_n)|+|f(x_n)-f_n(x_n)| by the triangle inequality. Can you see what to do from here?
 
  • #3
I got it now. thanks
 

What is the difference between convergence and continuity?

Convergence and continuity are two important concepts in mathematics. Convergence refers to the idea that a sequence of numbers or functions approaches a certain value or point as the number of terms in the sequence increases. On the other hand, continuity refers to the property of a function where small changes in the input result in small changes in the output. In other words, a function is continuous if it has no abrupt breaks or jumps in its graph.

How do you determine if a sequence or function is convergent?

To determine if a sequence or function is convergent, you need to look at its behavior as the number of terms or inputs increases. If the sequence or function approaches a fixed value or point, it is considered convergent. This can be shown mathematically by using a limit or by graphing the sequence or function and observing its behavior.

What is the formal definition of continuity?

The formal definition of continuity states that a function f(x) is continuous at a point c if the limit of f(x) as x approaches c exists and is equal to f(c). In simpler terms, this means that the value of the function at a specific point is equal to the limit of the function as it approaches that point.

Can a function be continuous but not convergent?

Yes, a function can be continuous but not convergent. This means that the function has no abrupt breaks or jumps in its graph, but it does not approach a fixed value or point as the input increases. An example of such a function is f(x) = sin(x), which is continuous but oscillates between -1 and 1 as x increases.

What is the importance of convergence and continuity in mathematics and science?

Convergence and continuity are important concepts in mathematics and science because they allow us to describe and understand the behavior of sequences and functions. They are used in a variety of fields, including calculus, analysis, and physics. For example, in physics, the concept of convergence is used to explain the behavior of objects in motion, while continuity is used to describe the smoothness of physical phenomena.

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