- #1
Nedeljko
- 40
- 0
Homework Statement
If [tex]f:R\longrightarrow R[/tex] is a infinitely differentiable function then the function [tex]g:R\longrightarrow R[/tex] defined as
[tex]
g(x)=\left\{
\begin{array}{ll}
\frac{f(x)-\sum_{k=0}^n\frac{f^{(k)}(0)}{k!}x^k}{x^{n+1}}, & x\neq 0,
\vspace{0.5em}\\
\frac{f^{(n+1)}(0)}{(n+1)!}, & x=0,
\end{array}
\right.
[/tex]
is also infinitely differentiable. Prove it.
Homework Equations
No.
The Attempt at a Solution
It is easy to prove that the function [tex]g[/tex] is continuous. By computing derivatives I can prove that the function is three times differentiable for example, but I can not make inductive step. Is this theorem known under any name?