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Can anyone make a suggestion how to deal with that problem and perhaps there are any books that will be useful for improving proficiency in the theme of uniform convergency and solving problems consequently. Thank you!enter image description here

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Since a continuous function on a closed and bounded interval is bounded and uniformly continuous, $I.$ and $II.$ are true ($I.$ is true by boundedness of $f$ and $II.$ is true by uniform continuity of $f$). Statement $III.$ is false -- you should be suspicious of it because there are continuous real functions on $[0,1]$ with unbounded derivative. Such functions would not satisfy $III.$. The answer is $(C)$.

I suggest you look in Bartle's real analysis text for problems on uniform convergence.

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$I$ has gotta hold. If $f$ is continuous, then the image of compact $[0,1]$ is compact, and therefore bounded. Let $C$ be the diameter of this bounded image.

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If $f\in C[0,1]$, then $f$ is bounded, so $I$ must hold. $III$ is wrong since $\sqrt[3]{x}$ is continuous on $[0,1]$ but is not Lipshitz here (which you may recognize is what $III$ says). $II$ also holds since all continuous functions on $[0,1]$ are uniformly continuous, and this is what $II$ says (with the arbitrary $\epsilon>0$ replaced with $1$ in the definition of uniform continuity).

Thus $c$ is the answer.

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