Continuous functions

Art of Problem Solving

AoPS Online

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

3 M G

BBookmark VNew Topic kLocked

College Math Topics in undergraduate and graduate studies

Topics in undergraduate and graduate studies

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Continuous functions

G H J

Reveal topic tags

real analysis unsolved

L

G H BBookmark kLocked kLocked NReply

Source: Romanian District Olympiad 2014, Grade 11, P2

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

836 posts

#1 h Jun 15, 2014, 2:27 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$g\colon\mathbb{R}\rightarrow\mathbb{R}$ , $g(x)=f(x)+f(2x)$ , and
$h\colon\mathbb{R}\rightarrow\mathbb{R}$ , $h(x)=f(x)+f(4x)$ , are continuous
functions. Prove that $f$ is also continuous.
Give an example of a discontinuous function $f\colon\mathbb{R} \rightarrow\mathbb{R}$ , with the following property: there exists an interval
$I\subset\mathbb{R}$ , such that, for any $a$ in $I$ , the function $g_{a} \colon\mathbb{R}\rightarrow\mathbb{R}$ , $g_{a}(x)=f(x)+f(ax)$ , is continuous.

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

1640 posts

aziiri

#2 h Jun 15, 2014, 2:52 PM • 2 Y

Y by Adventure10, Mango247

$g(2x)=f(2x)+f(4x)=g(x)-f(x)+h(x)-f(x)$ therefore : $f(x) =\frac{h(x)+g(x)-g(2x)}{4}$ , since $g,h$ are continuous we get that $f$ is continuous too.
I have a question for the second, is $f$ discontinuous at every point of $\mathbb{R}$ ?

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

15142 posts

#3 h Jun 15, 2014, 3:20 PM • 3 Y

Y by Adventure10, Mango247, and 1 other user

No, it is enough to exhibit an example where $f$ has just one point of discontinuity. Moreover, the interval $I$ must not be degenerated (to a point).

As an example, the signum function $f(0)=0$ and $f(x) = x/|x|$ for $x\neq 0$ , with $I=(-\infty,0)$ , for which $g_a$ is identically null for any $a\in I$ .

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

211 posts

#4 h Apr 7, 2025, 1:47 PM

Y by

Brothers and Sisters of AOPS, I am going to give my solution .....
Part 1:
We express $f(x)$ in terms of $g(x)$ and $h(x)$ and since $g(x)$ is continuous so $g(2x)$ is also continuous. So
$f(x)=\frac{g(x)+h(x)-g(2x)}{2}$ and hence $f(x)$ is continuous.
Part 2:
$f(x) = \begin{cases} 1, x\geq 0\\ -1, x<0 \end{cases}$

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a