Problems in Mathematics/To be added

< Problems in Mathematics

2 Exercise Suppose f is infinitely differentiable. Suppose, furthermore, that for every x, there is n such that f^{(n)}(x) = 0. Then f is a polynomial. (Hint: Baire's category theorem.)

Exercise e and \pi are irrational numbers. Moreover, e is neither an algebraic number nor p-adic number, yet e^p is a p-adic number for all p except for 2.

Exercise There exists a nonempty perfect subset of \mathbf R that contains no rational numbers. (Hint: Use the proof that e is irrational.)

Exercise Construct a sequence a_n of positive numbers such that \sum_{n \ge 1} a_n converges, yet \lim_{n \to \infty} {a_{n+1} \over a_n} does not exist.

Exercise Let a_n be a sequence of positive numbers. If \lim_{n \to \infty} n \left({a_n \over a_{n+1}} - 1 \right) > 1, then \sum_{n=1}^\infty a_n converges.

Exercise Prove that a convex function is continuous (Recall that a function f: (a,b) \rightarrow \mathbb{R} is a convex function if for all x,y \in (a,b) and all s,t \in [0,1] with s+t = 1, f(sx+ty) \leq sf(x)+tf(y))

Exercise Prove that every continuous function f which maps [0,1] into itself has at least one fixed point, that is \exists p \in [0,1] such that f(p) = p
Proof: Let g(x) = x - f(x). Then

Exercise Prove that the space of continuous functions on an interval has the cardinality of \mathbb{R}

Exercise Let f:[a,b] \rightarrow \mathbb{R} be a monotone function, i.e. \forall x,y \in [a,b]; x \leq y \Rightarrow f(x) \leq f(y). Prove that f has countably many points of discontinuity.

Exercise Suppose f is defined on the set of positive real numbers and has the property: f(xy) = f(x) + f(y). Then f is unique and is a logarithm.