1. ## real analysis question

this is from ross's book elementary analysis: the theory of calculus section 21 #10. i tried reading the section and stuff but it seems like the problems are still hard for me. here are the problems:

show that there exist continuous functions
(a) mapping (0,1) onto [0,1]
(b) mapping (0,1) onto R
(c) mapping [0,1]U[2,3] onto [0,1]

also, can someone give me tips on how i should study for this class? this class is the 2nd quarter of analysis i'm taking, and we are 4 weeks into the class. so i'd read the book and think i understand the material, only to get stumped on the questions... the midterm exam is on next week friday, so i'm getting really anxious... help!

2. Originally Posted by squarerootof2
this is from ross's book elementary analysis: the theory of calculus section 21 #10. i tried reading the section and stuff but it seems like the problems are still hard for me. here are the problems:

show that there exist continuous functions
(a) mapping (0,1) onto [0,1]
(b) mapping (0,1) onto R
(c) mapping [0,1]U[2,3] onto [0,1]
(a) Look at $\displaystyle \sin(2\pi x)$ this takes the interval $\displaystyle (0,1)$ to $\displaystyle [-1,1]$, so can you do something to modify this function so the image of $\displaystyle (0,1)$ is $\displaystyle [0,1]$?

RonL

3. Originally Posted by squarerootof2
this is from ross's book elementary analysis: the theory of calculus section 21 #10. i tried reading the section and stuff but it seems like the problems are still hard for me. here are the problems:

show that there exist continuous functions
(a) mapping (0,1) onto [0,1]
(b) mapping (0,1) onto R
(c) mapping [0,1]U[2,3] onto [0,1]

also, can someone give me tips on how i should study for this class? this class is the 2nd quarter of analysis i'm taking, and we are 4 weeks into the class. so i'd read the book and think i understand the material, only to get stumped on the questions... the midterm exam is on next week friday, so i'm getting really anxious... help!
For (b) consider somthing like $\displaystyle \tan( \pi x -\pi/2)$

RonL

4. Originally Posted by CaptainBlack
(a) Look at $\displaystyle \sin(2\pi x)$ this takes the interval $\displaystyle (0,1)$ to $\displaystyle [-1,1]$, so can you do something to modify this function so the image of $\displaystyle (0,1)$ is $\displaystyle [0,1]$?

RonL
isn't the image of the sin (2pi*x) function already [0,1]? it seems like all the pts in [0,1] are hit already by putting in (0,1) for x? i dont see how we need to modify the function. hmm...

5. Originally Posted by squarerootof2
isn't the image of the sin (2pi*x) function already [0,1]? it seems like all the pts in [0,1] are hit already by putting in (0,1) for x? i dont see how we need to modify the function. hmm...
try doing something like shrinking the period and shifting the graph

6. Originally Posted by squarerootof2

show that there exist continuous functions
(a) mapping (0,1) onto [0,1]
(b) mapping (0,1) onto R
(c) mapping [0,1]U[2,3] onto [0,1]
Actually we can construct ugly functions to satisfy requirements....
For (c) , we can get onto mapping with $\displaystyle \sin(\frac{\pi x}{2})$ from [0,1] to [0,1]. Now define f([2,3]) = 1. This will make the function continuous, but the function does not look natural

7. Originally Posted by squarerootof2
isn't the image of the sin (2pi*x) function already [0,1]? it seems like all the pts in [0,1] are hit already by putting in (0,1) for x? i dont see how we need to modify the function. hmm...
$\displaystyle \sin(2\pi x)$ achives a maximum value of $\displaystyle 1$ and a minimum of $\displaystyle -1$ in $\displaystyle (0,1)$ .

RonL

8. Originally Posted by Jhevon
try doing something like shrinking the period and shifting the graph
I would leave the period alone and shift the function upwards by 1 and shrink it by a factor of 2.

RonL

9. Originally Posted by CaptainBlack
$\displaystyle \sin(2\pi x)$ achives a maximum value of $\displaystyle 1$ and a minimum of $\displaystyle -1$ in $\displaystyle (0,1)$ .

RonL
so i've been thinking , and would it work if i take the absolute value of sin (2pi*x) (just to make the function onto [0,1]. would that work?

10. Originally Posted by squarerootof2
so i've been thinking , and would it work if i take the absolute value of sin (2pi*x) (just to make the function onto [0,1]. would that work?
1. It would leave the function continuous, so that's OK

2. It would map (0,1) to [0,1] so that's Ok

so yes it would

RonL