# function finding using continuity or derivative

I have a function $f:\mathbb{R}\rightarrow\mathbb{R}$ which is continuous at $\mathbb{R}$ and can derives at $\mathbb{R}^*$. Plus I have that $f(1)=1$ and $f'(1)=1$. Now I have: $$\frac{f(x)}{x^2}=\begin{cases}\ln{|x|}+\frac{1}{x^2}+c_1,x<0\\\ln{|x|}+\frac{1}{x^2}+c_2,x>0\end{cases}$$ and I want to find $c_1$ and $c_2$. I can easily find $c_2$, but how can I find $c_1$?

• You definitely want $c_1 = c_2$, that's easy to see. – Matti P. Apr 26 '18 at 10:30

With an arbitrary $c_1$ the function $f$ is continuous.

• I want to prove that $c_1=c_1$ – Leos Kotrop Apr 26 '18 at 10:38
• Prove $c_1=c_1$? :-O Sorry, but is it not always true? :) – Logic_Problem_42 Apr 26 '18 at 11:23