Continuous or Differentiable but Nowhere Lipschitz Continuous Function 
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*What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval?

*What is a real valued function that is differentiable on a close interval but not Lipschitz continuous on any subinterval?
 A: Continuous and nowhere Lipschitz
An example is given by the Weierstrass function, which is continuous and nowhere differentiable. This can be justified in two ways: 


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*A Lipschitz function is differentiable almost everywhere, by Rademacher's theorem.

*Direct inspection of the proof that the function is nowhere differentiable; the estimates used in the proof also imply it's nowhere Lipschitz ($|f(x+h)-f(x)|$ is estimated from below by $|h|^\alpha$ with $\alpha<1$, for certain $x,h$). 
Differentiable and nowhere Lipschitz
There are no such examples: a differentiable function on an interval must be Lipschitz on some subinterval. The following is an adaptation of a part of PhoemueX's answer.


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*The function $f'$ is a pointwise limit of continuous functions: namely, $$f'(x) = \lim_{n\to\infty} n(f(x+1/n)-f(x))$$
where for each $n$, the expression under the limit is continuous in $x$.

*Item 1 implies that $f'$ is continuous at some point $x_0$. This is a consequence of a theorem about functions of Baire class 1: see this answer for references.

*Continuity at $x_0$ implies $f'$ is bounded on some interval $(x_0-\delta,x_0+\delta)$. The Mean Value Theorem then implies that $f$ is Lipschitz on this interval.  
