Prove That Every Continuous F Can Be Written F g h
Problem 1
For which of the following functions $f$ is there a continuous function $F$ with domain $\mathbf{R}$ such that $F(x)=f(x)$ for all $x$ in the domain of $f ?$
(i) $\quad f(x)=\frac{x^{2}-4}{x-2}$
(ii) $\quad f(x)=\frac{|x|}{x}$
(iii) $\quad f(x)=0, x$ irrational.
(iv) $\quad f(x)=1 / q, x=p / q$ rational in lowest terms.
Carson Merrill
Numerade Educator
Problem 2
At which points are the functions of Problems $4-17$ and $4-19$ continuous?
Carson Merrill
Numerade Educator
Problem 3
(a) Suppose that $f$ is a function satisfying $|f(x)| \leq|x|$ for all $x$. Show that $f$ is continuous at $0 .$ (Notice that $f(0)$ must equal $0 .$ )
(b) Give an example of such a function $f$ which is not continuous at any $a \neq 0$
(c) Suppose that $g$ is continuous at 0 and $g(0)=0,$ and $|f(x)| \leq|g(x)|$ Prove that $f$ is continuous at 0
Carson Merrill
Numerade Educator
Problem 4
Give an example of a function $f$ such that $f$ is continuous nowhere, but $|f|$ is continuous everywhere.
Carson Merrill
Numerade Educator
Problem 5
For each number $a$. find a function which is continuous at $a$, but not at any other points.
Carson Merrill
Numerade Educator
Problem 6
(a) Find a function $f$ which is discontinuous at $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots .$ but continuous at all other points.
(b) Find a function $f$ which is discontinuous at $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots,$ and at $0,$ but continuous at all other points.
Problem 7
Suppose that $f$ satisfies $f(x+y)=f(x)+f(y),$ and that $f$ is continuous at $0 .$ Prove that $f$ is continuous at $a$ for all $a$.
Carson Merrill
Numerade Educator
Problem 8
Suppose that $f$ is continuous at $a$ and $f(a)=0 .$ Prove that if $\alpha \neq 0,$ then $f+\alpha$ is nonzero in some open interval containing $a$.
Carson Merrill
Numerade Educator
Problem 9
(a) Suppose $f$ is defined at $a$ but is not continuous at $a$. Prove that for some number $\varepsilon>0$ there are numbers $x$ arbitrarily close to $a$ with $|f(x)-f(a)|>\varepsilon .$ Illustrate graphically.
(b) Conclude that for some number $\varepsilon>0$ cither there are numbers $x$ arbitrarily close to $a$ with $f(x)<f(a)-\varepsilon$ or there are numbers $x$ arbitrarily close to $a$ with $f(x)>f(a)+\varepsilon$
Problem 10
(a) Prove that if $f$ is continuous at $a$, then so is $|f|$.
(b) Prove that every function $f$ continuous on $\mathrm{R}$ can be written $f=E+O$ where $E$ is even and continuous and $O$ is odd and continuous.
(c) Prove that if $f$ and $g$ are continuous, then so are $\max (f, g)$ and $\min (f, g)$
(d) Prove that every continuous $f$ can be written $f=g-h,$ where $g$ and $h$ are nonnegative and continuous.
Problem 11
Prove Theorem 1(3) by using Theorem 2 and continuity of the function $f(x)=1 / x$
Carson Merrill
Numerade Educator
Problem 12
(a) Prove that if $f$ is continuous at $l$ and $\lim _{x \rightarrow a} g(x)=l,$ then $\lim _{x \rightarrow a} f(g(x))=$ $f(l) .$ (You can go right back to the definitions, but it is easier to consider the function $G \text { with } G(x)=g(x) \text { for } x \neq a, \text { and } G(a)=1 .)$
(b) Show that if continuity of $f$ at $l$ is not assumed, then it is not generally true that $\lim _{x \rightarrow a} f(g(x))=f\left(\lim _{x \rightarrow a} g(x)\right) .$ Hint: Try $f(x)=0$ for $x \neq l,$ and
$f(l)=1$
Problem 13
(a) Prove that if $f$ is continuous on $[a, b],$ then there is a function $g$ which is continuous on $\mathbf{R},$ and which satisfies $g(x)=f(x)$ for all $x$ in $[a, b]$ Hint: since you obviously have a great deal of choice, try making $g$ constant on $(-\infty, a]$ and $[b, \infty)$
(b) Give an example to show that this assertion is false if $[a, b]$ is replaced
by $(a, b)$
Carson Merrill
Numerade Educator
Problem 14
(a) Suppose that $g$ and $h$ are continuous at $a,$ and that $g(a)=h(a) .$ Define $f(x)$ to be $g(x)$ if $x \geq a$ and $h(x)$ if $x \leq a .$ Prove that $f$ is continuous at $a$
(b) Suppose $g$ is continuous on $[a, b]$ and $h$ is continuous on $[b, c |$ and $g(b)=h(b) .$ Let $f(x)$ be $g(x)$ for $x$ in $[a, b]$ and $h(x)$ for $x$ in $[b, c]$ Show that $f$ is continuous on $[a, c] .$ (Thus, continuous functions can be "pasted together".)
Carson Merrill
Numerade Educator
Problem 15
Prove that if $f$ is continuous at $a$, then for any $\varepsilon>0$ there is a $\delta \geq 0$ so that whenever $|x-a|<\delta$ and $|y-a|<\delta,$ we have $|f(x)-f(y)|<\varepsilon$.
Problem 16
(a) Prove the following version of Theorem 3 for "right-hand continuity": Suppose that $\lim _{x \rightarrow a^{+}} f(x)=f(a),$ and $f(a)>0 .$ Then there is a number $\delta>0$ such that $f(x)>0$ for all $x$ satisfying $0 \leq x-a<\delta .$ Similarly, if $f(a)<0,$ then there is a number $\delta>0$ such that $f(x)<0$ for all $x$ satisfying $0 \leq x-a<\delta$
(b) Prove a version of Theorem 3 when $\lim _{x \rightarrow b^{-}} f(x)=f(b)$
Carson Merrill
Numerade Educator
Problem 17
If $\lim f(x)$ exists, but is $\neq f(a),$ then $f$ is said to have a removable discontinuity at $a$
(a) If $f(x)=\sin 1 / x$ for $x \neq 0$ and $f(0)=1,$ does $f$ have a removable discontinuity at $0 ?$ What if $f(x)=x \sin 1 / x$ for $x \neq 0,$ and $f(0)=1 ?$
(b) Suppose $f$ has a removable discontinuity at $a$. Let $g(x)=f(x)$ for $x \neq a,$ and let $g(a)=\lim _{x \rightarrow a} f(x) .$ Prove that $g$ is continuous at $a .$ (Don't work very hard; this is quite easy.)
(c) Let $f(x)=0$ if $x$ is irrational, and let $f(p / q)=1 / q$ if $p / q$ is in lowest terms. What is the function $g$ defined by $g(x)=\lim _{y \rightarrow x} f(y)$ ?
$*(\text { d) Let } f$ be a function with the property that every point of discontinuity is a removable discontinuity. This means that $\lim f(y)$ exists for all $x$
but $f$ may be discontinuous at some (even infinitely many) numbers $x$ Define $g(x)=\lim _{y \rightarrow x} f(y) .$ Prove that $g$ is continuous. (This is not quite so easy as part (b).)
Is there a function $f$ which is discontinuous at every point, and which has only removable discontinuities? (It is worth thinking about this problem now, but mainly as a test of intuition: even if you suspect the correct answer, you will almost certainly be unable to prove it at the present time. Sec Problem $22-33 .$ )
Source: https://www.numerade.com/books/chapter/continuous-functions/
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