Functions

Describe the sets of functions $f: \mathbb {R} \rightarrow \mathbb {R}$ possessing one of the following properties $\left(\varepsilon , \delta , x_{1}, x_{2} \in \mathbb {R}\right)$:

Let the function $f$ be defined on $\mathbb {R}$. Prove the equivalence of properties a)-e) of the function $f$:

1. the function $f$ is continuous on $\mathbb {R}$;

2. the set $f^{-1}(G)=\left\{ x \in \mathbb {R} \mid f(x) \in G\right\}$ is open for any open set $G \subset \mathbb {R}$;

3. the set $f^{-1}(F)=\left\{ x \in \mathbb {R} \mid f(x) \in F\right\}$ is closed for any closed set $F \subset \mathbb {R}$;

4. for any $c \in \mathbb {R}$ the sets $f^{-1}((-\infty , c))$ and $f^{-1}((c,+\infty ))$ are open;

5. for any $c \in \mathbb {R}$ the sets $f^{-1}((-\infty , c])$ and $f^{-1}([c,+\infty ))$ are closed.

6. Is the condition "for any $c \in \mathbb {R}$ the set $f^{-1}(\left\{ c\right\} )$ is closed" sufficient for the continuity of the function $f$?

Prove that any function defined on a set $E \subset \mathbb {R}$ has at most countably many points of discontinuity of the first kind.

Let $E_{L}, E_{R}$ be countable sets contained in $\left[0,1\right]$, $E_{L} \cap E_{R}=\varnothing$. Define on $\left[0,1\right]$ a function that is continuous only from the left at the points of $E_{L}$, continuous only from the right at the points of $E_{R}$, and continuous at the remaining points of the interval $\left[0,1\right]$.

Let $D_{L}(f)\left(D_{R}(f)\right)$ be the set of points at which a function defined on $\mathbb {R}$ is discontinuous from the left (respectively, from the right). Is it true that if one of these sets is at most countable, then the other is also at most countable?

Let $f_{0}$ be the Dirichlet function:

Prove that the set of discontinuity points of an arbitrary function defined on an interval $\Delta \subset \mathbb {R}$ is a set of type $F_{\sigma }$ (i.e., a union of a sequence of closed sets). Is this true if the function is defined on an arbitrary metric space?

Define on $\mathbb {R}$ a function with a given set of discontinuity points $E$, if:

Define on the interval $(0,1)$ a function such that any nonempty interval $\Delta \subset (0,1)$ contains a continuum of its points of continuity and points of discontinuity.

Prove that a function $f$ defined on an interval $\langle a, b\rangle$ is continuous if and only if

1. the function has the Cauchy property (i.e., the image of every interval $\langle p, q\rangle$ contained in $\langle a, b\rangle$ is an interval), and

2. the set $f^{-1}(\left\{ y\right\} )$ is closed for any $y \in \mathbb {R}$.

Prove that a function defined on a segment $\left[a, b\right]$ is continuous if and only if its graph is connected and closed.

Let $E \subset \mathbb {R}$, and let the function $f$ be defined on $E$ and have a local extremum at every non-isolated point of the set $E$. Prove that then the set $f(E)$ is at most countable. Is this true if $E$ is a separable metric space? an arbitrary metric space?

Describe all functions continuous on the segment $\left[a, b\right]$ that have a local extremum at every point of the interval $(a, b)$.

Let the function $f$ be defined and one-to-one on the interval $(a, b)$ and continuous at the point $c \in (a, b)$.

Let $f \in C(\langle a, b\rangle )$. Prove that the function $f$ is one-to-one if and only if it is strictly monotonic.

Let a positive function $f$ be defined and differentiable on the interval $[a,+\infty )$. Prove that if $\inf_{x \geqslant a} \frac{f^{\prime }(x)}{f(x)}>0$, then for any $\varepsilon >0$

Prove that if $f \in C([0,+\infty ))$ and for any $x>0$ the limit $\lim_{n \rightarrow \infty } f(n x)$ exists, then the limit $\lim_{x \rightarrow +\infty } f(x)$ exists.

Define on the interval $[0,+\infty )$ a function $f$ satisfying the conditions:

1. $\lim_{n \rightarrow \infty } f(n x)=0$ for any $x \in [0,+\infty )$;

2. $f$ is bounded on every finite interval and has only discontinuities of the first kind;

3. the set of partial limits of the function $f$ at infinity fills $\mathbb {R}$ (compare with problem 1.17).

Define on the interval $[0,+\infty )$ a function $f$ satisfying the conditions:

1. $\lim_{n \rightarrow \infty } f(n x)=0$ for any $x \in [0,+\infty )$;

2. the set of values of $f$ on any nonempty interval $\Delta \subset [0,+\infty )$ fills $\mathbb {R}$ (compare with problem 1.17).

Prove that if $f \in C([0,+\infty ))$ and $f(x+h)-f(x) \rightarrow 0$ as $x \rightarrow +\infty$ for any $h \in \mathbb {R}$, then $f(x+h)-f(x) \rightrightarrows 0$ as $x \rightarrow +\infty$ uniformly on any finite interval, and hence the function $f$ is uniformly continuous on $[0,+\infty )$.

Let $\delta >0, f \in C(\left[0,1\right])$. We say that the graph of $f$ has a horizontal chord of length $\delta$ if there is a point $x \in [0,1-\delta ]$ such that $f(x)=f(x+\delta )$. Prove that if $f(0)=f(1)$, then for any $n \in \mathbb {N}$ there is a horizontal chord of length $\frac{1}{n}$. Show that for chords of other lengths this is, in general, false.

Prove that if $f \in C(\mathbb {R})$ and for any $x \in \mathbb {R}$

then $f$ is a linear function.

We say that a function $f: \mathbb {R} \rightarrow \mathbb {R}$ is Cesàro continuous at the point $x_{0} \in \mathbb {R}$ if

Describe all functions that are Cesàro continuous at at least one point.

Let $f(x)=\prod_{k \geqslant 0}\left(1+x^{4^{k}}\right)$ for $0 \leqslant x<1$. Prove that

Does there exist a function continuous on the interval $\left[0,1\right]$ all of whose level sets are countable (equinumerous with $\mathbb {N}$)?

Does every positive polynomial in two real variables attain its infimum on the plane?

Let the function $f$ be defined and separately continuous on the square $Q=\left[0,1\right]^{2}$. Prove that there exists a sequence $f_{n}$ of functions continuous on $Q$ such that $f_{n}(x, y) \rightarrow f(x, y)$ for all $x, y \in [0,1]$ (i.e., $f$ is a function of the first Baire class on $Q$).

Let the function $f$ be defined and separately continuous on $\mathbb {R}^{2}$. Prove that if $f$ vanishes on an everywhere dense set (i.e., a set whose closure coincides with $\mathbb {R}^{2}$), then $f \equiv 0$.

Let the function $f$ be defined on $\mathbb {R}$. Prove that the following properties of the function $f$ are equivalent (compare with Problem 1.2):

1. the function $f$ is lower semicontinuous;

2. the set $f^{-1}((c,+\infty ))$ is open for every $c \in \mathbb {R}$;

3. the set $f^{-1}((-\infty , c])$ is closed for every $c \in \mathbb {R}$;

4. if $x_{n} \rightarrow a$, then $f(a) \leqslant \sup f\left(x_{n}\right)$;

5. for every point $a \in \mathbb {R}$ and every number $\varepsilon >0$ there exists a neighborhood $V$ of the point $a$ such that $f(x) \geqslant f(a)-\varepsilon$ for $x \in V$.

Prove that

Prove that if a function is lower semicontinuous on the segment $\left[a, b\right]$ and takes only finite values, then it attains its least value on $\left[a, b\right]$ (and hence is bounded below). Is it true that it is bounded above?

Let $g$ be an arbitrary function defined on an interval $\Delta \subset \mathbb {R}, f(x)=\varliminf_{y \rightarrow x} g(y)(x \in \Delta )$. Prove that the function $f$ is lower semicontinuous.

Let $\left\{ f_{\alpha }\right\}_{\alpha \in A}$ be a family of lower semicontinuous functions defined on an interval $\Delta \subset \mathbb {R}, f(x)=\sup_{\alpha \in A} f_{\alpha }(x)$ and $f(x)<+\infty$ for every $x \in \Delta$. Prove that the function $f$ is lower semicontinuous. Is this true for the function $h=\inf_{\alpha \in A} f_{\alpha }$? Will the function $h$ be lower semicontinuous if the family $\left\{ f_{\alpha }\right\}_{\alpha \in A}$ is finite? countable?

Prove that the function $f:\left[a, b\right] \rightarrow \mathbb {R}$ is lower semicontinuous if and only if

Let

Prove that a function $f \in C(\mathbb {R})$ is a polynomial of degree at most $m$ if and only if $\Delta_{h}^{m+1} f(x)=0$ for all $x, h \in \mathbb {R}$.

Let $f \in C^{1}([0,+\infty ))$. Prove that $f^{\prime }(x)+f(x) \rightarrow L \in \mathbb {R}$ as $x \rightarrow +\infty$ if and only if $f(x) \underset {x \rightarrow +\infty }{\longrightarrow } L$ and $f^{\prime }$ is uniformly continuous on $(0,+\infty )$.

Let the function $f$ be differentiable on the segment $\left[a, b\right]$. Prove that if $f^{\prime }(a) f^{\prime }(b)<0$, then there exists a point $c \in (a, b)$ such that $f^{\prime }(c)=0$ (Darboux's theorem).

Let the function $f$ be differentiable on the segment $\left[a, b\right]$. Prove that if the set $\left\{ x \in [a, b] \mid f^{\prime }(x)=y\right\}$ is closed for every $y \in \mathbb {R}$, then $f \in C^{1}(\left[a, b\right])$.

Let the function $f$ be defined on the interval $(a, b)$. Prove that $f \in C^{1}((a, b))$ if and only if the ratio $\frac{1}{h}(f(x+h)-f(x))$ tends, as $h \rightarrow 0$, to a finite limit uniformly on every interval $\left[\alpha , \beta \right]$ contained in ($a, b$).

Let $f \in C((a, b))$ and for every $x \in (a, b)$ let there exist a finite limit $\lim_{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}=g(x)$. Prove that

Let $f \in C((a, b))$ and for every $x \in (a, b)$ let there exist a finite limit $\lim_{h \rightarrow 0} \frac{f(x+h)+f(x-h)-2 f(x)}{h^{2}}=g(x)$. Prove that

Let the function $f$ be differentiable on ($a, b$) and satisfy the condition: for every point $x \in (a, b)$ there exists a number $\delta_{x}>0$ such that $f(x+h)-f(x-h)=2 h f^{\prime }(x)$ for $0<h<\delta_{x}$. Prove that the function $f$ is a polynomial of degree at most two.

Let $F \subset \mathbb {R}$ be a closed set with no interior points. Prove that there exists a strictly increasing function $f \in C^{1}(\mathbb {R})$ such that $\left\{ x \in \mathbb {R} \mid f^{\prime }(x)=0\right\} =F$.

Let $f \in C(\left[0,1\right])$. Are the statements equivalent:

1. the function $f$ is constant on $\left[0,1\right]$;

2. $\forall x \in (0,1) \exists \left\{ x_{n}\right\} : x_{n} \rightarrow x, x_{n}>x, \frac{f\left(x_{n}\right)-f(x)}{x_{n}-x} \rightarrow 0$;

3. $\forall x \in (0,1) \exists \left\{ x_{n}\right\} : x_{n} \rightarrow x, x_{n} \neq x, \frac{f\left(x_{n}\right)-f(x)}{x_{n}-x} \rightarrow 0$.

Can the derivative of a function that is everywhere differentiable and not constant on an interval vanish

Let $f \in V(\left[a, b\right]), \quad g(x)= =\stackrel{\boldsymbol {x}}{\underset {a}{\mathbf{V}}}(f) \quad (x \in (a, b]), g(a)=0$. Prove that if the function $f$ is continuous at the point $c \in [a, b]$, then the function $g$ is also continuous at that point.

Let $f \in C^{1}(\left[a, b\right])$. Prove that

Fig. 6

Prove that

Find the length of the graph of the Cantor function.

For which $\alpha >0$ does the Cantor function belong to the class $\operatorname {Lip}_{\alpha }(\left[0,1\right]) ?$

Let $K$ be a generalized Cantor set (see the definition before problem I.1.29), and let $\Delta_{\varepsilon_{1} \ldots \varepsilon_{n}}$ be its segments of rank $n$. Furthermore, let $a=\inf K, b=\sup K, \delta_{\varepsilon_{1} \ldots \varepsilon_{n}}$ be the intervals removed from the segments $\Delta_{\varepsilon_{1} \ldots \varepsilon_{n}}$ when constructing the segments of rank $(n+1)$.

Define on $\left[a, b\right]$ a function $\varphi$ as follows:

We call the function $\varphi$ the Cantor function corresponding to the set $K$. Prove that

Let $F \subset \mathbb {R}$ be a closed set with no isolated points. Prove that there exists a function $f \in C(\mathbb {R})$ such that

1. $f^{\prime }(x)=0$ for any $x \in \mathbb {R} \backslash F$;

2. $f(x+h)-f(x-h)>0$ for any $x \in F$ and any $h>0$.

Let $1<\alpha <2, f(x)=x^{2} \cos \frac{1}{x^{\alpha }}$ for $0<x \leqslant 1, f(0)=0$. Prove that $f \in V(\left[0,1\right])$, $f$ is differentiable on $\left[0,1\right]$, but the function $g(x)=\underset {0}{\stackrel{x}{\mathrm{~ V}}}(f)$ is not differentiable at zero. Does $g^{\prime }(0)$ exist for $\alpha =1$?

Let $\alpha , \beta >0, f(x)=x^{\alpha } \cos \frac{1}{x^{\beta }}$ for $0<x \leqslant 1, f(0)=0$. Prove that

Let $0<\gamma <1$. Construct a function $f$ of class $\operatorname {Lip}_{\gamma }(\left[0,1\right])$ that does not have bounded variation on any nondegenerate interval $\Delta \subset [0,1]$.

Exhibit functions $f, g \in C(\left[0,1\right])$ such that

Let the function $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuous and $2 \pi$-periodic,

Prove that $\varphi \in \operatorname {Lip}_{1 / 2}$, more precisely, that

Let $f(x)=\sum 2^{-n} \cos 4^{n} x$. Prove that $f \in \operatorname {Lip}_{1 / 2}$, but $f \notin \operatorname {Lip}_{\alpha }$ for $\alpha >\frac{1}{2}$.

Let $f(x)=\sum A^{-n} \sin 3^{n} x$, where $A \in (1,3)$. Prove that $f \in \operatorname {Lip}_{\alpha }$, where $\alpha =\log_{3} A$, but $f \notin \operatorname {Lip}_{\beta }$ for $\beta >\alpha$.

Let $f(x)=\sum 3^{-n} \sin 3^{n} x$. Prove that $f \in \operatorname {Lip}_{\alpha }$ for any $\alpha \in (0,1)$. Is it true that $f \in \operatorname {Lip}_{1}$?

Let $f(x)=\sum \frac{n!}{2^{n!}} \sin \left(2^{n!} \pi x\right)$. Prove that

Let $\omega$ be an arbitrary continuous increasing function defined on the half-line $[0, \infty )$, satisfying the conditions: $\omega (0)=0, \Omega (t)=\frac{\omega (t)}{t}$ decreases for $t>0$, and $\Omega (t) \underset {t \rightarrow +0}{\longrightarrow }+\infty$.

Prove that there exists a continuous, nowhere differentiable $2 \pi$-periodic function $F$ of the form