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

Jaqueline Kirby 2022-06-30 Answered
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.
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Answers (2)

Harold Cantrell
Answered 2022-07-01 Author has 21 answers
Any constant function f ( x ) = c will be an example. f is continuously differentiable, but f 1 can'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.

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Damon Stokes
Answered 2022-07-02 Author has 6 answers
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

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