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Differentiable Functions : Lipschitz and Absolute Values

I have a function that is differentiable on [a,b] and I am trying to figure out which scenario is more restrictive:

a) the function is a Lipschitz function with a Lipschitz constant L in (0,1)


b) the absolute value of f'(x) is less than one for all x in [a,b]

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Answer: (a) is more restrictive.
I claim that (a) can imply (b).
Suppose (a) holds. f(x) is a ...

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Differentiable Functions, Lipschitz and Absolute Values are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.