Consider a function defined on the closed interval such that is continuous and…
MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions
Problem
Consider a function f defined on the closed interval [a,b] such that f is continuous and strictly monotonic on that interval. Let R denote the range f([a,b]).
(a) Identify the precise condition that guarantees f possesses an inverse function on [a,b]. State the theorem on which your answer relies.
(b) Explain, using appropriate reasoning, why the continuity of f on [a,b] implies the continuity of f−1 on R. Your argument should reference a property of continuous functions on closed intervals.
(c) Assume further that f is differentiable on (a,b) with f′(x)=0 for all x in (a,b). Describe how this additional information affects the differentiability—and hence the continuity—of f−1 on R. Provide justification.