2.7.53 Function properties

Condition: Find a polynomial of the smallest degree \( g(x) \) such that the function \( f(x) \) is: 1) continuous on the entire line, 2) differentiable on the entire line if \[ f(x)=\left\{\begin{array}{ll} \frac{5 x}{4+x^{2}}, & |x| \geq 1 \\ g(x), & |x|<1 \end{array}\right. \]