Why are graphs of polynomials smooth and continuous?

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.