Continuity and Discontinuities of 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 $f \equiv 0$.