Zero limit

Let $f:[0, +\infty) \rightarrow \mathbb{R}$ be an integrable and uniformly continuous function. Prove that $\lim \limits_{x \rightarrow +\infty} f(x)=0$. Can the result hold if the function is just continuous instead of uniformly continuous? Give a brief explanation.

Solution

An $1-1$ function

Does there exist a function $f:\mathbb{R} \rightarrow \mathbb{R}$ with $f(\mathbb{R})=\mathbb{R}^*$ that is $1-1$ ?

Solution

What is the value of $f(0)$?

Let $f:[0, +\infty) \rightarrow \mathbb{R}$ be a twice differentiable function with a continuous second derivative. If

\begin{equation} \int_{0}^{\pi}\left ( f(x) \sin x + f''(x) \sin x \right )\, {\rm d}x =2 \end{equation}

and $f(\pi)=1$ hold , then evaluate $f(0)$.

Solution

Is it uniformly continuous?

Examine if the function $f(x)=x^2, \;\; x \in \bigcup \limits_{n=1}^{\infty}\left [ 2n, 2n+1 \right ]$ is uniformly continuous.

Solution

Existence of constants

Let $f:\mathbb{R} \rightarrow (0, +\infty)$ be a continuous function such that:

$$\int_0^1 f(x)\, {\rm d}x = \int_0^1 f^2(x)\, {\rm d}x $$

Prove that there exist $a, b \in \mathbb{R}$ such that $\displaystyle \int_a^b \frac{{\rm d}x}{f(x)}=1$.

Solution

There does not exist constant

Let $a_n$ be a sequence of positive numbers such that the series $\sum \limits_{n=1}^{\infty} a_n$ converges. Prove that there does not exist a positive real number $\ell$ such that:

$$\sum_{n=1}^{\infty}a_n \leq \ell \sum_{n=1}^{\infty}\frac{a_n}{\sum \limits_{k=1}^{n}\frac{1}{a_k}} $$

Solution

Continuous limit function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that the limit $\lim \limits_{y \rightarrow x} f(y)$ exists (and is in fact a real number) for all $x \in \mathbb{R}$.Let us call this limit $g(x)$. Prove that $g$ is continuous throughout $\mathbb{R}$.

Solution

Constant function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $\left | f(x)-f(y) \right |\leq \left | x -y \right |^2 , \;\; \forall x, y \in \mathbb{R}$. Prove that $f$ is constant.

Solution

Is completeness theorem satisfied?

Does the ordered field of the rational functions satisfy the completeness theorem: "all non - empty sets have a supremum")

Solution

Limit of nested integrals

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x) \geq 0$ forall $x$ and $\displaystyle \int_{-\infty}^{\infty} f(x)\, {\rm d}x=1$. For $r \geq 0$ , define:

$$I_n(r)= \idotsint \limits_{x_1^2+x_2^2+\cdots+x_n^2 \leq r} f(x_1) f(x_2) \cdots f(x_n) \;{\rm d}(x_1, x_2, \dots, x_n)$$

Evaluate the limit $\lim I_n(r)$ for a fixed $r$.

Solution

The identity function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a strictly increasing function in $\mathbb{R}$. If $f(r)=r, \; \forall r \in \mathbb{Q}$ prove that $f(x)=x , \; \forall x \in \mathbb{R}$.

Solution

Functions mapping an open interval to a closed one

Prove that there exist non constant functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that map any open interval onto a closed one.

Solution   $\newcommand{card}{{\rm card}}$

Does such function exist?

Let $0<a<1$. Does there exist a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:

$$\left|f(x)-f(y)\right |\geq \left|x-y\right|^a, \;\; \forall x, y \in \mathbb{R}$$

Solution

Inequality from Jensen

Let $f, g:\mathbb{R} \rightarrow \mathbb{R}$ be two continuous functions. If $f$ is convex then prove that:

$$f \left( \int_0^1 g(x) \, {\rm d}x \right) \leq \int_0^1 f\left(g(x)\right)\, {\rm d}x$$

Limit of a sequence

Find , if it exists, the limit of the sequence:

$$\mathop{\lim}\limits_{n\rightarrow{+\infty}}{({\underbrace{\sin\circ\sin\circ\ldots\circ\sin}_{n-\text{times}}})({n})}$$

Solution

Limit of a function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. If:

$$\lim_{x \rightarrow +\infty} [f(x)+f'(x)] =0$$

holds, then prove that $\displaystyle \lim_{x \rightarrow +\infty} f(x)=0$.

Solution

A function that is nowhere monotonic

Give an example of a function that is continuous but is nowhere monotonic. In continuity give an example of a function that is nowhere monotonic and has no extrema values.

Solution

Gronwall Inequality

Let $f, g$ be two continuous functions , non negative in $[a, b]$ and let $c>0$. If $\displaystyle f(x) \leq c + \int_a^b f(t) g(t) \, {\rm d}t$ then prove that:

$$f(x) \leq ce^{\displaystyle \int_a^x g(t)\, {\rm d}t}$$

Solution

Let us copy ... Fourier

Let $a_n$ be a strictly increasing sequence of positive integers. Prove that

$$x_n =\frac{1}{a_1} + \frac{1}{a_1 a_2} +\cdots + \frac{1}{a_1 a_2 a_3 \cdots a_n}$$

converges to an irrational number.

Solution

Riemann integrability

Let $f:[0, 1] \rightarrow \mathbb{R}$ be defined as:

$$f(x)= \left\{\begin{matrix}
 0&,  &x \in [0,1]\cap \left ( \mathbb{R} \setminus \mathbb{Q} \right ) \\
x_n &,  &x=q_n  \in [0,1] \cap \mathbb{Q} \\
\end{matrix}\right.$$

where $x_n$ is a sequence such that $\lim x_n =0$ and $0\leq x_n \leq 1$ and $q_n$ is an enumeration of the rationals of the interval $[0, 1]$.

Prove that $f$ is Riemann integrable and that $\displaystyle \int_0^1 f(x)\, {\rm d}x =0$.

Solution