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c) If $ f $ is differentiable on $ (a,b) $ and $ \lim\limits_{x \rightarrow a+}f(x) $ exists and is finite, then for each $ x \in (a,b) $ there is a $ c $ between $ a $ and $ x $ such that $ f(x) - f(a+) = f'(c)(x-a). $

d) If $ f $ and $ g $ are differentiable on $ [a,b] $ and $ |f'(x)| \leq 1 \leq |g'(x)| $ for all $ x \in (a,b), $ then $ |f(x) - f(a)| \leq |g(x) - g(a)| $ for all $ x \in [a,b] $.

Could someone provide hints on how to begin both proofs? I apologize if these are trivial. I am having a difficult time making the connection between the derivative of a function, cord functions, and the Mean Value Theorem.

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    what means $f(a+)$? – 2017-02-28

2 Answers 2

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c) If we define $f(a):=f(a+)$, then f is continuous on $[a,b)$. The assertion follows from the mean value theorem.

d) Use the generalized mean value theorem.

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    If you don't mind, could you give me a reference on partially extending intervals from open to closed? I never covered this in class. – 2017-02-28
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consider $y=x$, from $(0,1)$ and $y = 5-x$ from$(\infty,0]$ as counterexample for first case. A hint for the second problem: $g(x)$ is monotonic and use the bounds of derivative.