Smooth function has derivatives of all orders. Polynomials have derivatives of all orders, thus polynomials are smooth functions.
For example: f(x)=2x+3 => f'(x)=2 => f''(x)=0 => f'''(x)=0...
So all derivatives exist. (Derivative being zero is ok.)
Their continuity can be proven using the Weierstrass (epsilon-delta) definition.