These problems are real analysis related problems. One of these is Liptuaz continuity/holder condition. It would be nice if you use mean value theorem for solving that problem.
1. Let f : [0, 2] ? R be continuous, assume that f is twice differentiable at all points of (a, b), and assume that f(0) = 0, f(1) = 1 and f(2) = 2. Prove: There exists c ? (0, 2) such that f??(c) = 0.
2. Let f : (a, b) ? R, where a, b ? R and a < b and suppose f is monotone. Prove lim x?c+ f(x) and lim x?c? f(x) exist at all c ? (a, b).
3. Let f : [0, 1] ? R be continuous, differentiable at all points of (0, 1). Assume f?(x) ? 16 for all x ? (0, 1). Prove there is some interval J ? [0, 1] of length 1/4 such that |f(x)| ? 4 for all x ? J.
4. Let I be an open interval in R, f : I ? R and suppose that f satisfies the following condition: There exist constants C and ?, C > 0 and ? > 1, such that |f(x) ? f(y)| <C|x ? y|?for all x, y ? I. Prove: f is constant. (A function that satisfies a Holder condition of order ? with ? > 1 is a constant.)
This solution clearly assesses real analysis. Differentiability in real analysis are solved.