Explain in detail thank you

Problem 3. Suppose f E CO (R") and let S = f-({0}). If the gradient Vf(x) = (if(x), . .. . Onf(I) ) does not equal the zero vector for any r E S, then show that S is a submanifold of R" of dimension n - 1. Use this result to prove that the n-sphere S" C R"+ is a n-dimensional submanifold of Ra+1