Inverse functions

We prove the inverse function theorem. A continuously differentiable function which has non-zero derivative at a point has an inverse in a neighbourhood of that point (see Figure 6).

Figure 6: The Inverse Function

A function is called monotone if it preserves or reverses order over the domain of its definition. In other words if (f (x) - f (y))(x - y) does not change sign for all x and y in the domain of definition of f. We say f is strictly monotone if in addition the above product is zero only when x = y. It is clear that a strictly monotone function is one-to-one.

Exercise 46   Let f be monotone and continuous in for x in the interval a < x < b. Show that there is a function g on the range of f so that g(f (x)) = x for all x in the interval a < x < b. (Hint: Use the intermediate value theorem and extremal value theorem).

One way to ensure that a function is monotone is to use Rolle's theorem in reverse to prove:

Exercise 47   If f is continuously differentiable and f'(x₀) $\neq$ 0 then show that f is monotonic in some interval around x₀. Hence show that f has a inverse g (as in the exercise above) in some small enough interval around f (x₀).

We can also compute the formal inverse

Exercise 48   If f can be expressed as

f (x) = f (x₀) + f₁(x - x₀) + ... + f_n(x - x₀)ⁿ + o((x - x₀)ⁿ)

with f₁ $\neq$ 0, then show that the inverse function g(y) has the following form where y₀ = f (x₀).

g(y) = x₀ + $${\frac{1}{f_1}}$$(y - y₀) - $${\frac{f_2}{f_1^3}}$$(y - y₀)² + ... + g_n(y - y₀)ⁿ + o((y - y₀)ⁿ)

where g_n is of the form P_n(f₁,..., f_n)/f₁^{n + 1}, where P_n is a polynomial function.