Problems in Mathematics/To be added

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.