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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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    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.2012-12-20

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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.