Bonus problem Let $\varphi:\mathbb R\to\mathbb R$ be a function. Prove that $\varphi$ is convex in the sense of Exercise 1 if and only if, for any $x,y\in\mathbb R$ and $\theta\in[0,1]$, we have $$ \varphi(\theta x+(1-\theta)y)\le \theta \varphi(x)+(1-\theta)\varphi(y). $$ Show also that a convex function is continuous, possesses left and right derivative at every point, and is differentiable on the complement of a countable subsets.

Bonus problem Is it possible to construct a function $\varphi:\mathbb R\to\mathbb R$ that is differentiable outside a countable set $N\subset\mathbb R$ such that $\varphi'(x)=0$ for $x\notin N$ and $\varphi(x)\to\pm\infty$ as $x\to\pm\infty$ ?

Bonus problem Let $\varphi:[0,1]\to\mathbb R$ be a differentiable function. Is it true that the derivative $\varphi':[0,1]\to\mathbb R$ is integrable in the sense of Riemann and $$ \varphi(1)-\varphi(0)=\int_0^1\varphi'(y)\,dy\, ? $$

Let $\varphi:[0,1]\to\mathbb R$ be a differentiable function such that the derivative $\varphi':[0,1]\to\mathbb R$ is bounded. Prove that $$ \varphi(x)=\varphi(0)+\int_0^x\varphi'(y)\,dy, \quad x\in[0,1]. $$

Tasks for June 29: resolve Exercises 1-5

ReplyDeleteBonus problem

Let $\varphi:\mathbb R\to\mathbb R$ be a function. Prove that $\varphi$ is convex in the sense of Exercise 1 if and only if, for any $x,y\in\mathbb R$ and $\theta\in[0,1]$, we have

$$

\varphi(\theta x+(1-\theta)y)\le \theta \varphi(x)+(1-\theta)\varphi(y).

$$

Show also that a convex function is continuous, possesses left and right derivative at every point, and is differentiable on the complement of a countable subsets.

Tasks for June 30: resolve Exercises 6-9

ReplyDeleteBonus problem

Is it possible to construct a function $\varphi:\mathbb R\to\mathbb R$ that is differentiable outside a countable set $N\subset\mathbb R$ such that $\varphi'(x)=0$ for $x\notin N$ and $\varphi(x)\to\pm\infty$ as $x\to\pm\infty$ ?

Tasks for July 1: resolve Exercises 10-13

ReplyDeleteBonus problem

Let $\varphi:[0,1]\to\mathbb R$ be a differentiable function. Is it true that the derivative $\varphi':[0,1]\to\mathbb R$ is integrable in the sense of Riemann and

$$

\varphi(1)-\varphi(0)=\int_0^1\varphi'(y)\,dy\, ?

$$

This comment has been removed by the author.

DeleteCan we say for opposing example $$\phi(x)=x^{\frac{1}{3}}?$$

DeleteThis function is not differentiable at $x=0$.

DeleteThis comment has been removed by the author.

DeleteIndeed, may be \[

Deletef(x)=

\begin{cases}

$x^2\sin(\frac{1}{x^2}),& x\neq 0$\\

0, & x=0

\end{cases}

\]

Yes, this is a counterexample showing that the property is not true.

DeleteTasks for July 2: resolve Exercises 14-18

ReplyDeleteLet $\varphi:[0,1]\to\mathbb R$ be a differentiable function such that the derivative $\varphi':[0,1]\to\mathbb R$ is bounded. Prove that

$$

\varphi(x)=\varphi(0)+\int_0^x\varphi'(y)\,dy, \quad x\in[0,1].

$$

Tasks for July 8: resolve Exercises 22-24 and finish the bonus problem of June 29

ReplyDeleteTasks for July 9: resolve Exercises 25-27

ReplyDeleteTasks for July 12 & 13: revision of the first 27 exercises

ReplyDeleteTasks for July 14: complete the revision of the first 27 exercises

ReplyDeleteTasks for July 15: resolve Exercises 28-30 and write down their solutions

ReplyDeleteTasks for July 16: resolve Exercises 31-35 and write down their solutions

ReplyDeleteTasks for July 19: resolve Exercises 36-38 and write down their solutions

ReplyDeleteTasks for July 21: resolve Exercises 39-42 and write down their solutions

ReplyDeleteTasks for July 22: resolve Exercises 43-45 and write down their solutions

ReplyDeleteTasks for July 22: resolve Exercises 46-48 and write down their solutions

ReplyDelete