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I attempted some easy multiple-choice and T/F questions to test if I am entirely clear about the topic before I do any proofy works. It's essential to be clear on these basic concepts and ideas.

Let me know whether I am right or wrong and I'll appreciate if you briefly explain:) Thanks for helping!

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I am not sure about 1. But I think it's true.

  1. a.Yes, for sure b.no??, but I am not sure, yet I can't tell the difference btw this and a... c. Yes d. yes. But again, not sure
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for 1: write down the definition of $f$ being continuous at the point 3. For 2: What do you know about continuity on a closed set? – Alex Dec 20 '12 at 0:31
You are not ready to write proofs, but the impressionistic, train-of-thought presentation of your thought process was good. – gnometorule Dec 20 '12 at 0:38
@Alex For 2. I know f is bounded because continuous function on a compact interval is compact. right ? – Victoria J. Dec 20 '12 at 0:43
It's the image $f([a, b])$ that's compact (compactness is a property of sets, not functions), and we know compact subsets of $\mathbb{R}^n$ are closed and bounded. – Brian Dec 20 '12 at 2:01

1 Answer

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As a counterexample for 2.(d), consider the function $$f(x)=\left|x-\frac12\right|.$$ This is continuous on $[0,1]$, but fails to be differentiable at $\frac12$.

For 2.(b), there's a nice theorem that says if a function is continuous on a compact interval, then it's uniformly continuous there.

Both 2.(a) and 2.(c) are fine. Follow Alex's advice for 1.

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