Differentiable function/D in R/Inverse function/Fact/Proof

Proof

We consider the difference quotient

${}{\frac {f^{-1}(y)-f^{-1}(b)}{y-b}}={\frac {f^{-1}(y)-a}{y-b}}\,,$

and have to show that the limit for ${}y\rightarrow b$ exists, and obtains the value claimed. For this, let ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ denote a sequence in ${}E\setminus \{b\}$, converging to ${}b$. Because of fact, the function ${}f^{-1}$ is continuous. Therefore, also the sequence with the members ${}x_{n}:=f^{-1}(y_{n})$ converges to ${}a$. Because of bijectivity, ${}x_{n}\neq a$ for all ${}n$. Thus

${}\lim _{n\rightarrow \infty }{\frac {f^{-1}(y_{n})-a}{y_{n}-b}}=\lim _{n\rightarrow \infty }{\frac {x_{n}-a}{f(x_{n})-f(a)}}={\left(\lim _{n\rightarrow \infty }{\frac {f(x_{n})-f(a)}{x_{n}-a}}\right)}^{-1}\,,$

where the right-hand side exists, due to the condition, and the second equation follows from fact.

To fact