Proof
We consider the difference quotient
-
![{\displaystyle {}{\frac {f^{-1}(y)-f^{-1}(b)}{y-b}}={\frac {f^{-1}(y)-a}{y-b}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed8d757793b5b372a13a0fcf877d2fc6bf74910)
and have to show that the limit for
exists, and obtains the value claimed. For this, let
denote a
sequence
in
,
converging
to
. Because of
fact,
the function
is continuous. Therefore, also the sequence with the members
converges to
. Because of bijectivity,
for all
. Thus
-
![{\displaystyle {}\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}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184cb2113a09d5cae0e1c8168950f5da340bfa50)
where the right-hand side exists, due to the condition, and the second equation follows from
fact.