# 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.