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 $\sum _{n\geq 1}a_{n}$ 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.