(1). We write
and
respectively with the objects which were formulated in
fact,
that is
-
![{\displaystyle {}f(x)=f(a)+s(x-a)+r(x)(x-a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d8e5719f01bb2908781b5780aa174e849b9b26)
and
-
![{\displaystyle {}g(x)=g(a)+{\tilde {s}}(x-a)+{\tilde {r}}(x)(x-a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac5b93507d98218f295fecb53223b2aaadd4985)
Summing up yields
-
![{\displaystyle {}f(x)+g(x)=f(a)+g(a)+(s+{\tilde {s}})(x-a)+(r+{\tilde {r}})(x)(x-a)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7522ba6ea67c3df4ef0835140a90119cf2eed00)
Here, the sum
is again continuous in
, with value
.
(2). We start again with
-
![{\displaystyle {}f(x)=f(a)+s(x-a)+r(x)(x-a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d8e5719f01bb2908781b5780aa174e849b9b26)
and
-
![{\displaystyle {}g(x)=g(a)+{\tilde {s}}(x-a)+{\tilde {r}}(x)(x-a)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55391241c5e3b580f2ff7d3944307c1c55107f56)
and multiply both equations. This yields
![{\displaystyle {}{\begin{aligned}f(x)g(x)&=(f(a)+s(x-a)+r(x)(x-a))(g(a)+{\tilde {s}}(x-a)+{\tilde {r}}(x)(x-a))\\&=f(a)g(a)+(sg(a)+{\tilde {s}}f(a))(x-a)\\&\,\,\,\,\,+(f(a){\tilde {r}}(x)+g(a)r(x)+s{\tilde {s}}(x-a)+s{\tilde {r}}(x)(x-a)+{\tilde {s}}r(x)(x-a)+r(x){\tilde {r}}(x)(x-a))(x-a).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a4ea48f054b326c6cfcffc8e21ee372507e29e)
Due to
fact
for
limits,
the expression consisting of the last six summands is a continuous function, with value
for
.
(3) follows from (2), since a constant function is differentiable with derivative
.
(4). We have
-
![{\displaystyle {}{\frac {{\frac {1}{g(x)}}-{\frac {1}{g(a)}}}{x-a}}={\frac {-1}{g(a)g(x)}}\cdot {\frac {g(x)-g(a)}{x-a}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93e4307c07b326b8368f49478d01f96fbbbba6b0)
Since
is continuous in
, due to
fact,
the left-hand factor converges for
to
, and because of the differentiability of
in
, the right-hand factor converges to
.
(5) follows from (2) and (4).