A paraboloid is a surface. $$z=x^2+y^2$$ gives paraboloid. Let $\sigma$ be a surface patch for a paraboloid defined by $$\sigma (u,v)= (u,v, u^2+v^2)$$ I want to show that this is a surface. To show that $\sigma S is 1-1 continuous with inverse continuous is enough to prove that this is a surface. Right? Or using atlas, should I prove this question? If I need to use atlas, how?

You are right, proving that $\sigma$ is continuous with continuous inverse is sufficient. Also $\\{ \sigma \\}$ gives an atlas consisting of only one chart.