Uniform Continuity Homework: Showing Limits and Restrictions

[chief12]

Homework Statement

1)Show, if E is a subset of D is a subset of the real numbers R and f maps D into R is uniformly continuous, then the restriction of f to E is also uniformly continuous.

2)Show, if f is continuous and real valued on [a,b) and if the limit of f(x) as x approaches b exists, then f is uniformly continuous.

Homework Equations

The Attempt at a Solution

1)since E[tex]\subseteq[/tex]D, then any value in E is also in D, therefore if f is continuous on D, it must be continuous on d.

2) since a function is f:(a,b)---> R is uniformally continious on (a,b) iff f can be extended continuously to [a,b]. Since the limit exists, then the interval can be changed to [a,b], as f(b) has a definite value.

 

[mighty2000]

Therefore, f is uniformly continuous on [a,b].

Thank you for your post. I am a scientist and I would like to provide some feedback on your attempted solutions.

1) Your explanation for the first problem is not entirely clear. It is correct that if E is a subset of D, then any value in E is also in D. However, this does not necessarily mean that if f is continuous on D, it must be continuous on E. The key here is that f maps D into R, so it is possible for f to be continuous on D but not on E. To show that the restriction of f to E is also uniformly continuous, you can use the definition of uniform continuity and the fact that E is a subset of D.

2) Your explanation for the second problem is also not entirely clear. It is true that a function f:(a,b) -> R is uniformly continuous on (a,b) if and only if it can be extended continuously to [a,b]. However, the fact that the limit of f(x) as x approaches b exists does not automatically mean that f can be extended continuously to [a,b]. To show that f is uniformly continuous on [a,b], you can use the definition of uniform continuity and the fact that f is continuous on [a,b).

I hope this helps clarify your solutions. Keep up the good work!