function finding using continuity or derivative

Question

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 $c_1$?

Answer 1

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