Continuity and liimit of functions

In summary, the conversation discusses whether a given f_n function is necessarily continuous based on the fact that lim_{n \rightarrow \infty}f_n(x) exists for each x in [0,1] and is denoted by f(x). It is concluded that this fact alone is not enough to prove that f is continuous, and a counter-example must be constructed to show that f may not be continuous.
  • #1
jeff1evesque
312
0

Homework Statement


Suppose [tex]f_n : [0, 1]\rightarrow R[/tex] is continuous and lim[tex]_{n \rightarrow \infty}f_n(x)[/tex] exists for each x in [0,1]. Denote the limit by [tex]f(x)[/tex].

Is f necessarily continuous?

Homework Equations


We know by Arzela-Ascoli theorem:
If [tex]f_n: [a,b] \rightarrow R[/tex] is continuous, and [tex]f_n[/tex] converges to [tex]f [/tex]uniformly, then [tex]f[/tex] is continuous.

The Attempt at a Solution


Question: Does the fact of knowing
lim[tex]_{n \rightarrow \infty}f_n(x)[/tex] exists for each [tex]x \in [0,1][/tex]. Denote the limit by f(x).
give us insight to declare that [tex]f_n[/tex] converges to [tex]f[/tex] uniformly- and thus satisfying Arzela-Ascoli's theorem?

Thanks,Jeffrey Levesque
 
Last edited:
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  • #2
Well, no. They only gave you that the fn are continuous and converge pointwise. I think they want you find an example of a sequence of continuous functions that are pointwise convergent but don't have an continuous limit. Can you think of one?
 
  • #3
Can someone provide some insight for me as to what the following means:

jeff1evesque said:
lim[tex]_{n \rightarrow \infty}f_n(x)[/tex] exists for each x in [0,1]. Denote the limit by [tex]f(x)[/tex].

And how I could use this fact to construct my justification for whether f is necessarily continuous?
 
  • #4
f isn't necessarily continuous. Face it. It says fn converges at each point. That's not enough to prove f is continuous.
 
  • #5
Dick is suggesting that you find a counter-example. Taking fn to be piecewise linear will suffice.
 
  • #6
The Arzela-Ascoli theorem asserts something about a sequence of equicontinuous functions. This has little to do with your question [or you have seen a different version of A-A].

Just construct a counter-example, i.e. a sequence of continuous functions (f_n)_n which converges pointwise to some discontinuous function. (Every book that introduces the concept of 'uniform convergence' will have such a counter-example, so you probably have encountered one already.)
 

Related to Continuity and liimit of functions

What is the definition of continuity of a function?

The continuity of a function at a given point means that the function has a well-defined value at that point and that the values of the function near that point do not change drastically. In other words, the function is continuous if it can be drawn without lifting the pencil from the paper.

What is the difference between continuity and differentiability?

Continuity and differentiability are closely related concepts, but they are not the same. A function is continuous if it has a well-defined value at a given point and the values of the function near that point do not change drastically. On the other hand, a function is differentiable if it has a well-defined derivative at a given point, which means that the function is smooth and has no sharp corners or breaks.

How do you determine if a function is continuous at a specific point?

To determine if a function is continuous at a specific point, you need to check if the function has a well-defined value at that point and if the limit of the function as it approaches that point exists and is equal to the function value. If both of these conditions are met, then the function is continuous at that point.

What are the three types of discontinuities?

The three types of discontinuities are removable, jump, and infinite discontinuities. A removable discontinuity occurs when there is a hole in the graph of the function, but the function can be made continuous by filling in the hole. A jump discontinuity occurs when the function has two distinct values at a point, and an infinite discontinuity occurs when the limit of the function at a point is either positive or negative infinity.

What is the definition of a limit of a function?

The limit of a function at a given point is the value that the function approaches as the input approaches that point. In other words, it is the value that the function gets closer and closer to as the input gets closer and closer to the given point.

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