What Are Some Examples of Unique Real Analysis Functions?

gordon_0451
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Homework Statement


Please give examples

-functions continuous nowhere, continuous at one point

– functions differentiable everywhere but with discontinuous derivative

– Examples of uniformly continuous functions, functions not uniformly continuous

– Combinations of the above. For example, find a function, f : [0, 1] ! [0, 1], having the
intermediate-value property but continuous at only one point

thank you very much!


Homework Equations


Rolle’s theorem
Mean-value theorem
Cauchy mean-value theorem


The Attempt at a Solution



I have trouble to start it . Please help.
 
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hi gordon_0451, welcome to PF - you need to show your attempts & people will try & help
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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