Is There a Nowhere Dense Continuous Function on [0,1]?

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The discussion focuses on demonstrating the existence of a continuous function f: [0,1] to R that is not Lipschitz on any subinterval [r,s] using the Baire Category Theorem. The user has defined the set A(r,s) as the collection of continuous functions that are Lipschitz on the interval [r,s] and has established that A(r,s) is closed. However, the user is struggling to prove that this set is nowhere dense, which is crucial for applying the Baire Category Theorem effectively.

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itzik26
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hey,
I need to show, using Baire Category Theorem, that there exits a continuous function
f: [0,1] to R , that isn't Lipschitz on the interval [r,s] for every 0<=r<s<=1 .

I defined the set A(r,s) to be all the continuous functions that are lipschitz on the interval [r,s]. I showed that A(r,s) is closed , but I'm having trouble showing it's nowhere dense.

help please! :)
 
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Would you you let us know the version of Baire's thm. you are trying? Also, which topology /metric did you use?
 

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