Introduction to Continuity

Abstract

In this chapter the algebra of continuous functions is established. A function that is continuous at each irrational number and discontinuous at each rational number is constructed. This function is know as the Riemann function, the Thomae function, the ruler function, or the raindrop function.

Appendices

Problems

1.1 Problems for Sect. 2.1

• 1. If $g$ is continuous at $L$ and $f(x)\to L$ as $x\to a$, prove $g(f(x))\to g(L)$ as $x\to a$.

• 2. Let $f:[0,1]\to [0,1]$ be determined by $f(0)=0,$ and for any $n\in \mathbb {N}$, $f(x)=1/n$, when $\frac {1}{n+1}<x\leq \frac {1}{n}$. Since $\bigcup _{n=1}^{\infty }\left ]\frac {1}{n+1},\frac {1}{n}\right ]=]0,1]$ and the union is disjoint, $f$ is a function defined on the closed interval $[0,1]$.

  • a. Prove that $f$ is increasing, i.e., $x<y\implies f(x)\leq f(y)$.

  • b. Prove that $f$ is continuous at $0$.

  • c. Prove that $f$ is continuous at $1$.

  • d. Prove that $f$ is continuous on $\left ]\frac {1}{n+1},\frac {1}{n}\right [$ for all $n\in \mathbb {N}$.

  • e. Prove that $f$ is discontinuous at $x=1/n$ for all $n\in \mathbb {N}$ with $n\geq 2$.

• 3. Let $D:=[-1,1]\cup \{3\}\cup [5,7]$ and let $f:D\to \mathbb {R}.$ Then $f$ is continuous at $3.$

• 4. If $f:\mathbb {R}\to \mathbb {C}$ is continuous on $\mathbb {R}$ and $f(x)=x^{2}$ for every rational $x$, show $f(x)=x^{2}$ for every real $x$.

• 5. If $f:\mathbb {R}\to \mathbb {R}$ is continuous and

  $$ f(x+y)=f(x)+f(y)\text{ for all }x,y\in\mathbb{R}, $$

  then there is constant $c\in \mathbb {R}$, such that $f(x)=cx$, for all $x$ in $\mathbb {R}$. [Hint: $f(2)=f(1+1)=2f(1)$, and $f(1)=f(1/2)+f(1/2)=2f(1/2)$, so $f(1/2)=\frac {1}{2}f(1)$].

• 6. Let $f:\mathbb {C}\to \mathbb {C}$ be continuous at $a$. Suppose $\left (x_{n}\right )$ is a sequence of complex numbers converging to $a$. Prove the sequence $\left (f\left (x_{n}\right )\right )$ converges to $f\left (a\right )$.

• 7. Why does the composition rule for limits (Theorem 1.4.14) not imply the composition rule for continuity (Theorem 2.1.1)

1.2 Problems for Sect. 2.2

• 1. Let $\sigma$ be the pseudo-sine function. Let $f(x):=\sigma (1/x)$, when $x\neq 0$ and let $f(0):=0$. Show that $f$ is discontinuous at $0$.

• 2. Let $\sigma$ be the pseudo-sine function. Let $g(x):=x\sigma (1/x)$ for $x\neq 0$. Prove $g$ has a removable discontinuity at $0$.

1.3 Problems for Sect. 2.3

• 1. Prove the the function in Problem 2 for Sect. 2.1 is continuous from the left at every point in the half-open interval $]0,1]$.

Solutions and Hints for the Exercises

Exercise 2.1.6. For a fixed $q$, there are only finitely many $p$ such that $a-\gamma$ $\leq p/q$ $\leq a+\gamma$. Alternatively, for any integer $k\geq 1$, the two integers closest to $ka$ are $\left \lfloor ka\right \rfloor$ and $\left \lfloor ka\right \rfloor +1$, in fact $\left \lfloor ka\right \rfloor <ka<\left \lfloor ka\right \rfloor +1$. Hence, the largest $\gamma$ satisfying the desired conclusion is the smallest of the numbers $a-\frac {\left \lfloor ka\right \rfloor }{k}$, $\frac {\left \lfloor ka\right \rfloor +1}{k}-a$, $k=1,2,\ldots ,M$.

Exercise 2.1.8. This is a consequence of Exercise 2.1.6 and Corollary 1.4.20.

Exercise 2.2.1. Let $L:=\lim _{x\to a}f(x)$ exists, then

$$ g(x):=\begin{cases} f(x) & \text{when }x\neq a\\ L & \text{when }x=a \end{cases} $$

is continuous at $a$.

Exercise 2.3.2. Similar to the corresponding result for one-sided limits.