Define: \(\displaystyle \lim_{x\to a-}f(x)=-\infty\), \(\displaystyle \lim_{x\to-\infty}f(x)=b\)- and \(\displaystyle \lim_{x\to-\infty}f(x)=+\infty\).

Difficulty: 1.

Formulate the negation of \(\displaystyle \lim_{x\to a}f(x)=+\infty\)!

Difficulty: 1.

Prove that the function \(\displaystyle [x]\) is continuous in \(\displaystyle a\) if \(\displaystyle a\) is not an integer, and left-continuous if \(\displaystyle a\) is an integer.

Difficulty: 1.

Find a monotone function \(\displaystyle f:[0,1]\to[0,1]\) with infinitely many points of discontinuity.

Difficulty: 1.

Formulate the definition using the letters \(\displaystyle \varepsilon,\delta,P,Q\) etc.:

\(\displaystyle \lim_{-\infty}f=1; \quad \lim_{t\to t_0+0}s(t)=0; \quad \lim_{\zeta\to -0}g(\zeta)=-\infty \quad \Olim_{\vartheta\to-1}h(\vartheta)=\infty; \quad \Ulim_{\xi\to-\infty}u(\xi)=2. \)

Difficulty: 1.

Formulate the definition using the letters \(\displaystyle \varepsilon,\delta,K,L\) etc.

\(\displaystyle \lim_{1}f=\infty; \quad \lim_{\eta\to\eta_0-}s(\eta)=2; \quad \lim_{x\to \infty}g(x)=-\infty; \quad \lim_{\omega\to\omega_0-}s(\omega)=2; \quad \Ulim_{0+}g=1; \quad \Ulim_{\infty}h=1. \)

Difficulty: 1.

Find a good \(\displaystyle \delta\) or \(\displaystyle L\) for \(\displaystyle \varepsilon >0\) or for \(\displaystyle K\) for the following functions.

 (1)  \(\displaystyle \lim\limits_{x\to 1+}(x^2+1)/(x-1),\)      (2)  \(\displaystyle \lim\limits_{x\to \infty}\frac{\sin(x)}{\sqrt x}\).

Difficulty: 2.

Determine the points of discontinuity of the following functions. What type of discontinuities are these?

 (1)  \(\displaystyle f(x)=\frac{x^{3}-1}{x-1},\)      (2)  \(\displaystyle g(x)=\frac{x^{2}-1}{|x-1|},\)      (3)  \(\displaystyle h_{1}(x)=x[\frac{1}{x}],\)      (4)  \(\displaystyle h_{2}(x)=x^{2}[\frac{1}{x}].\)

Difficulty: 2.

Determine the points of discontinuity of the following functions. What type of discontinuities are these?
a) \(\displaystyle f(x)=\frac{x-2}{x^{2}-x-2},\)    b) \(\displaystyle g(x)=\sgn \left (\left \{ \frac{1}{x} \right \} \right )\).

Difficulty: 2.

Prove that \(\displaystyle \lim_{x\to a}f(x)=b\iff\lim_{x\to a-0}f(x)=\lim_{x\to a+0}f(x)=b\).

Difficulty: 2.

In which points are the following functions continuous?

 (1)  \(\displaystyle f(x)=\begin{cases} {x} & \text{if $\frac1x\in \N$} \\ 0 & \text{if $\frac1x\not\in \N$;} \end{cases}\)      (2)  \(\displaystyle f(x)=\begin{cases} 3x+7 & \text{if $x\in \Q$} \\ 4x & \text{if $x\not\in \Q$}; \end{cases}\)      (3)  \(\displaystyle f(x)=\begin{cases} x^2 & \text{if $x\geq0$} \\ cx & \text{if $x<0$}. \end{cases}\)

Difficulty: 2.

Where are they continuous?

 (1)  Riemann-function,      (2)  \(\displaystyle \sin\dfrac1x\),      (3)  \(\displaystyle x\sin\dfrac1x\).

Difficulty: 2.

Prove that if \(\displaystyle f:\R \to \R\) and \(\displaystyle g:\R \to \R\) are continuous and \(\displaystyle f(a)<g(a)\), then \(\displaystyle a\) has a neighborhood, where \(\displaystyle f(x)<g(x)\).

Difficulty: 2.

Let \(\displaystyle f\) be convex in \(\displaystyle (-\infty,\infty)\) and assume that \(\displaystyle \lim\limits_{x\to-\infty}f(x)=\infty.\) Is it possible that \(\displaystyle \lim\limits_{x\to\infty}f(x)=-\infty?\)

Difficulty: 2.

Let \(\displaystyle f\) be convex in \(\displaystyle (-\infty,\infty)\) and assume that \(\displaystyle \lim\limits_{x\to-\infty}f(x)=0.\) Is it possible that \(\displaystyle \lim\limits_{x\to\infty}f(x)=-\infty?\)

Difficulty: 2.

Prove that if \(\displaystyle f\) and \(\displaystyle g\) are continuous in the point \(\displaystyle a\) then \(\displaystyle \max(f,g)\) and \(\displaystyle \min(f,g)\) are also continuous in the point \(\displaystyle a\).

Difficulty: 2.

Does the continuity of \(\displaystyle g(x)=f(x^2)\) imply the continuity of \(\displaystyle f(x)\)?

Difficulty: 2.

Can we extend \(\displaystyle \frac{\sqrt{x} -1}{x-1}\) to \(\displaystyle x=1\) continuously?

Difficulty: 2.

Prove that a function \(\displaystyle f:\R\to\R\) is continuous if and only if the preimage of every open set is open.

Difficulty: 2.

In which points is the following function continuous?

\(\displaystyle f(x)=\begin{cases} \sin {1\over x} & \text{if $x\neq0$} \\ 0 & \text{if $x=0$} \end{cases}\)

Difficulty: 2.

In which points is the following function continuous?

\(\displaystyle f(x)=\begin{cases} x\sin {1\over x} & \text{if $x\neq0$} \\ 0 & \text{if $x=0$} \end{cases}\)

Difficulty: 2.

In which points is the following function continuous?

\(\displaystyle f(x)=\begin{cases} e^{-{1\over x^2}} & \text{if $x\neq0$} \\ 0 & \text{if $x=0$} \end{cases}\)

Difficulty: 2.

\(\displaystyle \Ulim_{x\to\infty} \Big(\{2x\}^2-4\{x\}^2\Big)=? \qquad \Olim_{x\to\infty} \Big(\{2x\}^2-4\{x\}^2\Big)=? \)

Difficulty: 2.

Determine the points of discontinuity of the following functions. What type of discontinuities are these?

 (1)  \(\displaystyle \frac{x^3-1}{(x-1)(x-2)(x-3)}\),      (2)  \(\displaystyle \frac1{\left[\frac1x\right]}\).

Difficulty: 3.

The continuity of the function \(\displaystyle f:\R \to \R\) at the point \(\displaystyle a\) is defined by:

\(\displaystyle (\forall \varepsilon >0 ) (\exists \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) .\)

Consider the following variations of this formula.

\(\displaystyle (\forall \varepsilon >0 ) (\forall \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\exists \varepsilon >0 ) (\forall \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\exists \varepsilon >0 ) (\exists \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\forall \delta >0 ) (\exists \varepsilon >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\exists \delta >0 ) (\forall \varepsilon >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) .\)

Which properties of \(\displaystyle f\) are described by these formulas?

Difficulty: 3.

Find an \(\displaystyle f\) and \(\displaystyle g\) such that \(\displaystyle \lim\limits_{x\to \alpha}f(x)=\beta\), \(\displaystyle \lim\limits_{x\to \beta}g(x)=\gamma\), but \(\displaystyle \lim\limits_{x\to \alpha}g(f(x))\neq\gamma\).

Difficulty: 3.

Prove that if \(\displaystyle f:\R \to \R\) is periodic and \(\displaystyle \lim_{x\to \infty} f(x)=0,\) then \(\displaystyle f\) is identically zero.

Difficulty: 3.

Prove that if \(\displaystyle f:[0,1]\to\R\) is continuous, then \(\displaystyle g(x):=\min\{f(x),0\}\) is also continuous.

Difficulty: 3.

Assume that \(\displaystyle g(x)=\lim\limits_{t\to x}f(t)\) exists in every point. Prove that \(\displaystyle g(x)\) is continuous.

Difficulty: 7.

Prove that if a function \(\displaystyle \R\to\R\) is continuous in every rational point, then there is an irrational point as well where it is continuous.

Difficulty: 7.

Is there an \(\displaystyle \R\to\R\) function for which the limit is \(\displaystyle \infty\) at every point?

Difficulty: 7.

Suppose that the function \(\displaystyle f:\R\to\R\) is continuous, and \(\displaystyle f(n\cdot a)\to0\) for all \(\displaystyle a>0\). Prove that \(\displaystyle \lim\limits_{x\to\infty} f=0\).

Difficulty: 8.

What is the cardinality of the set of continuous \(\displaystyle \R\to\R\) functions?

Difficulty: 8.