Chapter 3

Using the definition of continuity directly prove that \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x):=x^{2}\) is continuous.

Suppose \(f:[0,1] \rightarrow \mathbb{R}\) is monotone. Prove \(f\) is bounded.

Let \(f: S \rightarrow \mathbb{R}\) be uniformly continuous. Let \(A \subset S .\) Then the restriction \(\left.f\right|_{A}\) is uniformly continuous.

Find an example of a discontinuous function \(f:[0,1] \rightarrow \mathbb{R}\) where the conclusion of the intermediate value theorem fails.

Find the limit or prove that the limit does not exist a) \(\lim _{x \rightarrow c} \sqrt{x},\) for \(c \geq 0\) b) \(\lim _{x \rightarrow c} x^{2}+x+1,\) for any \(c \in \mathbb{R}\) c) \(\lim _{x \rightarrow 0} x^{2} \cos (1 / x)\) d) \(\lim _{x \rightarrow 0} \sin (1 / x) \cos (1 / x)\) e) \(\lim _{x \rightarrow 0} \sin (x) \cos (1 / x)\)

Using the definition of continuity directly prove that \(f:(0, \infty) \rightarrow \mathbb{R}\) defined by \(f(x):=1 / x\) is continuous.

Find an example of a bounded discontinuous function \(f:[0,1] \rightarrow \mathbb{R}\) that has neither an absolute minimum nor an absolute maximum.

Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be a function. Define \(g:(0,1] \rightarrow \mathbb{R}\) via \(g(x):=f(1 / x) .\) Using the definitions of limits directly, show that \(\lim _{x \rightarrow 0^{+}} g(x)\) exists if and only if \(\lim _{x \rightarrow \infty} f(x)\) exists, in which case they are equal.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by $$ f(x):=\left\\{\begin{array}{ll} x & \text { if } x \text { is rational, } \\ x^{2} & \text { if } x \text { is irrational } \end{array}\right. $$ Using the definition of continuity directly prove that \(f\) is continuous at 1 and discontinuous at 2 .

Show that \(f:(c, \infty) \rightarrow \mathbb{R}\) for some \(c>0\) and defined by \(f(x):=1 / x\) is Lipschitz continuous.

Let \(f:(0,1) \rightarrow \mathbb{R}\) be a continuous function such that \(\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 1} f(x)=0 .\) Show that \(f\) achieves either an absolute minimum or an absolute maximum on (0,1) (but perhaps not both).

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by $$ f(x):=\left\\{\begin{array}{ll} \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Is \(f\) continuous? Prove your assertion.

Show that \(f:(0, \infty) \rightarrow \mathbb{R}\) defined by \(f(x):=1 / x\) is not Lipschitz continuous.

Let $$ f(x):=\left\\{\begin{array}{ll} \sin (1 / x) & \text { if } x \neq 0, \\ 0 & \text { if } x=0. \end{array}\right. $$ Show that \(f\) has the intermediate value property. That is, for any \(ay>f(b),\) then there exists a \(c \in(a, b)\) such that \(f(c)=y\).

Let us justify terminology. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(\lim _{x \rightarrow \infty} f(x)=\infty\) (diverges to infinity). Show that \(f(x)\) diverges (i.e. does not comverge) as \(x \rightarrow \infty\),

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by $$ f(x):=\left\\{\begin{array}{ll} x \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Is \(f\) continuous? Prove your assertion.

Let \(A, B\) be intervals. Let \(f: A \rightarrow \mathbb{R}\) and \(g: B \rightarrow \mathbb{R}\) be uniformly continuous functions such that \(f(x)=g(x)\) for \(x \in A \cap B\). Define the function \(h: A \cup B \rightarrow \mathbb{R}\) by \(h(x):=f(x)\) if \(x \in A\) and \(h(x):=g(x)\) if \(x \in B \backslash A\) a) Prove that if \(A \cap B \neq \emptyset,\) then \(h\) is uniformly continuous. b) Find an example where \(A \cap B=\emptyset\) and \(h\) is not even continuous.

Suppose \(g(x)\) is a monic polynomial of odd degree \(d\), that is, $$ g(x)=x^{d}+b_{d-1} x^{d-1}+\cdots+b_{1} x+b_{0}, $$ for some real numbers \(b_{0}, b_{1}, \ldots, b_{d-1} .\) Show that there exists \(a K \in \mathbb{N}\) such that \(g(-K)<0 .\) Hint: Make sure to use the fact that \(d\) is odd. You will have to use that \((-n)^{d}=-\left(n^{d}\right)\).

Come up with the definitions for limits of \(f(x)\) going to \(-\infty\) as \(x \rightarrow \infty, x \rightarrow-\infty,\) and as \(x \rightarrow c\) for a finite \(c \in \mathbb{R}\). Then state the definitions for limits of \(f(x)\) going to \(\infty\) as \(x \rightarrow-\infty,\) and as \(x \rightarrow c\) for a finite \(c \in \mathbb{R}\)

Suppose \(S \subset \mathbb{R},\) and \(f: S \rightarrow \mathbb{R}\) is an increasing function. Prove: a) If \(c\) is a cluster point of \(S \cap(c, \infty)\), then \(\lim _{x \rightarrow c^{+}} f(x)<\infty\). b) If \(c\) is a cluster point of \(S \cap(-\infty, c)\) and \(\lim f(x)=\infty,\) then \(S \subset(-\infty, c)\).

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a polynomial of degree \(d \geq 2 .\) Show that \(f\) is not Lipschitz continuous.

Suppose \(g(x)\) is a monic polynomial of positive even degree \(d,\) that is, $$ g(x)=x^{d}+b_{d-1} x^{d-1}+\cdots+b_{1} x+b_{0}, $$ for some real numbers \(b_{0}, b_{1}, \ldots, b_{d-1} .\) Suppose \(g(0)<0 .\) Show that \(g\) has at least two distinct real roots.

Suppose \(P(x):=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\) is a monic polynomial of degree \(n \geq 1\) (monic means that the coefficient of \(x^{n}\) is \(\left.1\right) .\) a) Show that if \(n\) is even, then \(\lim _{x \rightarrow \infty} P(x)=\lim _{x \rightarrow-\infty} P(x)=\infty\), b) Show that if \(n\) is odd, then \(\lim _{x \rightarrow \infty} P(x)=\infty\) and \(\lim _{x \rightarrow-\infty} P(x)=-\infty(\) see previous exercise).

Prove the following statement. Let \(S \subset \mathbb{R}\) and \(A \subset S .\) Let \(f: S \rightarrow \mathbb{R}\) be a continuous function. Then the restriction \(\left.f\right|_{A}\) is continuous.

Suppose I \(\subset \mathbb{R}\) is an interval and \(f: I \rightarrow \mathbb{R}\) is a function such that for each \(c \in I,\) there exist \(a, b \in \mathbb{R}\) with \(a>0\) such that \(f(x) \geq a x+b\) for all \(x \in I\) and \(f(c)=a c+b .\) Show that \(f\) is strictly increasing.

Let \(f:(0,1) \rightarrow \mathbb{R}\) be a bounded continuous function. Show that the function \(g(x):=\) \(x(1-x) f(x)\) is uniformly continuous.

Let \(\left\\{x_{n}\right\\}\) be a sequence. Consider \(S:=\mathbb{N} \subset \mathbb{R},\) and \(f: S \rightarrow \mathbb{R}\) defined by \(f(n):=x_{n} .\) Show that the two notions of limit, $$ \lim _{n \rightarrow \infty} x_{n} \quad \text { and } \quad \lim _{x \rightarrow \infty} f(x) $$ are equivalent. That is, show that if one exists so does the other one, and in this case they are equal.

Find an example of a function \(f:[-1,1] \rightarrow \mathbb{R}\) such that for \(A:=[0,1],\) the restriction \(\left.f\right|_{A}(x) \rightarrow 0\) as \(x \rightarrow 0\), but the limit of \(f(x)\) as \(x \rightarrow 0\) does not exist. Note why you cannot apply Proposition \(3.1 .15 .\)

Suppose \(f: I \rightarrow J\) is a continuous, bijective (one-to-one and onto) function for two intervals I and J. Show that \(f\) is strictly monotone.

Show that \(f:(0, \infty) \rightarrow \mathbb{R}\) defined by \(f(x):=\sin (1 / x)\) is not uniformly continuous.

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and periodic with period \(P>0\). That is, \(f(x+P)=f(x)\) for all \(x \in \mathbb{R}\). Show that \(f\) achieves an absolute minimum and an absolute maximum.

Find example functions \(f\) and \(g\) such that the limit of neither \(f(x)\) nor \(g(x)\) exists as \(x \rightarrow 0\), but such that the limit of \(f(x)+g(x)\) exists as \(x \rightarrow 0\).

Give an example of functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) such that the function h defined by \(h(x):=f(x)+g(x)\) is continuous, but \(f\) and \(g\) are not continuous. Can you find \(f\) and \(g\) that are nowhere continuous, but \(h\) is a continuous function?

Consider a monotone function \(f: I \rightarrow \mathbb{R}\) on an interval I. Prove that there exists a function \(g: I \rightarrow \mathbb{R}\) such that \(\lim _{x \rightarrow c^{-}} g(x)=g(c)\) for all \(c \in I,\) except the smaller (left) endpoint of \(I,\) and such that \(g(x)=f(x)\) for all but countably many \(x .\)

Let \(f: \mathbb{Q} \rightarrow \mathbb{R}\) be a uniformly continuous function. Show that there exists a uniformly continuous function \(\tilde{f}: \mathbb{R} \rightarrow \mathbb{R}\) such that \(f(x)=\widetilde{f}(x)\) for all \(x \in \mathbb{Q}\).

(Challenging): Suppose \(f(x)\) is a bounded polynomial, in other words, there is an \(M\) such that \(|f(x)| \leq M\) for all \(x \in \mathbb{R}\). Prove that \(f\) must be a constant.

Let \(c_{1}\) be a cluster point of \(A \subset \mathbb{R}\) and \(c_{2}\) be a cluster point of \(B \subset \mathbb{R}\). Suppose \(f: A \rightarrow B\) and \(g: B \rightarrow \mathbb{R}\) are functions such that \(f(x) \rightarrow c_{2}\) as \(x \rightarrow c_{1}\) and \(g(y) \rightarrow L\) as \(y \rightarrow c_{2}\). If \(c_{2} \in B\), also suppose that \(g\left(c_{2}\right)=L\). Let \(h(x):=g(f(x))\) and show \(h(x) \rightarrow\) Las \(x \rightarrow c_{1} .\) Hint: Note that \(f(x)\) could equal \(c_{2}\) for many \(x \in A\), see also Exercise 3.1 .14

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be continuous functions. Suppose that for all rational numbers \(r, f(r)=g(r)\). Show that \(f(x)=g(x)\) for all \(x\).

a) Find a continuous \(f:(0,1) \rightarrow \mathbb{R}\) and a sequence \(\left\\{x_{n}\right\\}\) in (0,1) that is Cauchy, but such that \(\left\\{f\left(x_{n}\right)\right\\}\) is not Cauchy. b) Prove that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, and \(\left\\{x_{n}\right\\}\) is Cauchy, then \(\left\\{f\left(x_{n}\right)\right\\}\) is Cauchy.

Suppose \(f:[0,1] \rightarrow[0,1]\) is continuous. Show that \(f\) has a fixed point, in other words, show that there exists an \(x \in[0,1]\) such that \(f(x)=x\).

Let \(c\) be a cluster point of \(A \subset \mathbb{R}\), and \(f: A \rightarrow \mathbb{R}\) be a function. Suppose for every sequence \(\left\\{x_{n}\right\\}\) in \(A\), such that \(\lim x_{n}=c\), the sequence \(\left\\{f\left(x_{n}\right)\right\\}_{n=1}^{\infty}\) is Cauchy. Prove that \(\lim _{x \rightarrow c} f(x)\) exists.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be continuous. Suppose \(f(c)>0 .\) Show that there exists an \(\alpha>0\) such that for all \(x \in(c-\alpha, c+\alpha)\) we have \(f(x)>0\)

(Challenging): Find an example of an increasing function \(f:[0,1] \rightarrow \mathbb{R}\) that has a discontinuity at each rational number. Then show that the image \(f([0,1])\) contains no interval. Hint: Enumerate the rational numbers and define the function with a series.

Prove: a) If \(f: S \rightarrow \mathbb{R}\) and \(g: S \rightarrow \mathbb{R}\) are uniformly continuous, then \(h: S \rightarrow \mathbb{R}\) given by \(h(x):=f(x)+g(x)\) is uniformly continuous. b) If \(f: S \rightarrow \mathbb{R}\) is uniformly continuous and \(a \in \mathbb{R},\) then \(h: S \rightarrow \mathbb{R}\) given by \(h(x):=a f(x)\) is uniformly continuous.

Find an example of a continuous bounded function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that does not achieve an absolute minimum nor an absolute maximum on \(\mathbb{R}\).

Suppose I is an interval and \(f: I \rightarrow \mathbb{R}\) is monotone. Show that \(\mathbb{R} \backslash f(I)\) is a countable union of disjoint intervals.

Prove: a) If \(f: S \rightarrow \mathbb{R}\) and \(g: S \rightarrow \mathbb{R}\) are Lipschitz, then \(h: S \rightarrow \mathbb{R}\) given by \(h(x):=f(x)+g(x)\) is Lipschitz. b) If \(f: S \rightarrow \mathbb{R}\) is Lipschitz and \(a \in \mathbb{R},\) then \(h: S \rightarrow \mathbb{R}\) given by \(h(x):=a f(x)\) is Lipschitz.

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function such that \(x \leq f(x) \leq x+1\) for all \(x \in \mathbb{R}\). Find \(f(\mathbb{R})\).

Let \(f: S \rightarrow \mathbb{R}\) be a function and \(c \in S\), such that for every sequence \(\left\\{x_{n}\right\\}\) in \(S\) with \(\lim x_{n}=c\), the sequence \(\left\\{f\left(x_{n}\right)\right\\}\) converges. Show that \(f\) is continuous at \(c\).

Suppose \(f:[0,1] \rightarrow(0,1)\) is increasing. Show that for any \(\varepsilon>0,\) there exists a strictly increasing \(g:[0,1] \rightarrow(0,1)\) such that \(g(0)=f(0), f(x) \leq g(x)\) for all \(x,\) and \(g(1)-f(1)<\varepsilon\).