$\displaystyle g(x)=\left\{\begin{array}{cc}x^2+1,&\mbox{ if }x\leq 0\\x^2+bx\sin(\frac{1}{x})+c,&\mbox{ if }x>0\end{array}\right.$
$\displaystyle b\in$
$\displaystyle c\in$
2. The only possible point of discontinuity is at $\displaystyle x=0,$ hence, you must put $\displaystyle \lim_{x\to0^-}f(x)=\lim_{x\to0^+}f(x).$