# Problems in Mathematics/To be added

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

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

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

Exercise Construct a sequence ${\displaystyle a_{n}}$ of positive numbers such that ${\displaystyle \sum _{n\geq 1}a_{n}}$ converges, yet ${\displaystyle \lim _{n\to \infty }{a_{n+1} \over a_{n}}}$ does not exist.

Exercise Let ${\displaystyle a_{n}}$ be a sequence of positive numbers. If ${\displaystyle \lim _{n\to \infty }n\left({a_{n} \over a_{n+1}}-1\right)>1}$, then ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ converges.

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

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

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

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

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