Yet another proof function continuity related

[bluevires]

http://www.uAlberta.ca/~blu2/question1.gif


hey guys, I've tried this question and here's what I come up with, however I don't think this is anywhere near the right answer, but it does show the direction that I'm trying to work toward, I would appreciate any tips/help on how should I approach this question.

http://www.uAlberta.ca/~blu2/answer.gif

 

[AKG]

Let e be an arbitrary positive real number. Since f is continuous at L, there is a positive real d such that |L-x|< d implies |f(L)-f(x)| < e. Since the limit of g at a is L, then there exists a positive real c such that |x-a| < c implies |g(x)-L| < d. Is this enought of a hint?

 

[Architeuthis Dux]

I understand your approach and appreciate your effort in trying to solve this problem. However, I believe there may be some misunderstandings about the concept of continuity and how to prove it. Continuity is a fundamental concept in mathematics and is closely related to the concept of limits. In order to prove that a function f(x) is continuous at a point x=a, we need to show that the limit of f(x) as x approaches a is equal to the value of f(a). In other words, f(x) must be defined and have a unique value at x=a, and the limit of f(x) as x approaches a must exist and be equal to f(a).

Looking at the provided graph, we can see that the function is defined and has a unique value at x=1, which is f(1)=2. However, in order to prove continuity, we also need to show that the limit of f(x) as x approaches 1 exists and is equal to 2.

To do this, we can use the definition of a limit and show that for any given epsilon (ε), we can find a corresponding delta (δ) such that for all x within δ of 1, the difference between f(x) and 2 (the value of the limit) is less than epsilon. This would prove that the limit of f(x) as x approaches 1 is indeed 2, and therefore the function is continuous at x=1.

In summary, proving continuity involves showing that a function is defined and has a unique value at a point, and that the limit of the function at that point exists and is equal to the value of the function. I suggest revisiting the definition of continuity and limits, and using that as a guide to approach this question. Keep up the good work!