Find an example of a function f : [ a , b ] → R which is continuous, but not strictly increasing, such that no inverse function f − 1 satisfy the property of the Inverse Function Theorem.

Question

Find an example of a function   f  :  [  a  ,  b  ]  →  R which is continuous, but not strictly increasing, such that no inverse function       f          −      1       satisfy the property of the Inverse Function Theorem.

Harold Cantrell · Accepted Answer

Any constant function   f  (  x  )  =  c will be an example.   f is continuously differentiable, but       f          −      1       can&#039;t be a function as you would have       f          −      1        (  c  )  =  x and       f          −      1        (  c  )  =      x    ′   where   x  ≠      x    ′  , thereby violating the definition of a function.

Damon Stokes · Accepted Answer

Consider the function   f  (  x  )  =      x    2    sin  ⁡  (      1    x    ) defined on   [  0  ,  1  ]. Then you can see that       f    ′    (  0  )  =  1 but       f    ′   fails to be continuously differentiable as       f    ′   is not continuous at   0. And also you can see that there is no neighborhood of   0 such that       f          −      1       exists

Find an example of a function f : [ a , b ] → R which is contin

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