Proving Continuity in Functions: A Comparison of Two Statements

[stukbv]

Homework Statement

1. if f : [-1,1] --> Reals is such that sin(f(x) is continuous on the reals then f is continuous.

2. if f : [-1,1] --> Reals is such that f(sin(x)) is continuous on the reals then f is continuous.

Are these true or false how do i prove / give a counter example?

 

[HallsofIvy]

Again "If you do not show at least some attempt to do this problem yourself, this thread will be deleted."

 

[stukbv]

ok, its just i have no idea where to start really.
I know that the composition of 2 continuous functions is continuous too so I think that the second one is true, since sinx is continuous everywhere, the only way the composition could be discontinuous is is f was discontinuous??

 

[micromass]

Hi stukbv!

Take your favorite discontinuous function and try it out!

 

[stukbv]

are you talking about number 1 or 2? I think i have disproved number 1 so that's okay.. now just 2 :(

 

[micromass]

are you talking about number 1 or 2? I think i have disproved number 1 so that's okay.. now just 2 :(

Try to prove that one. What definition of continuity would you like to use?

 

[stukbv]

e-d i think ,

so for |x-c| < d we have that |f(sin(x))-f(sin(c))| < e

 

[micromass]

So for our epsilon we need to find a delta such that

$$|x-c|<\delta~\Rightarrow~|f(x)-f(c)|<\varepsilon$$

Now, the trick is, can you write that x and c as sines of something close together?

 

[stukbv]

i don't know what you mean

 

[micromass]

Can you write x=sin(y) and c=sin(d)?

 

[stukbv]

oh ok so we get |sin(y)-sin(d)| < delta => |f(sin(y))-f(sin(d))| < epsilon?

 

[micromass]

oh ok so we get |sin(y)-sin(d)| < delta => |f(sin(y))-f(sin(d))| < epsilon?

Yes, and if you know that |y-d| is small, then |f(sin(y))-f(sin(d))| is also small, by hypothesis...

 

[stukbv]

so is that it then ?

 

[micromass]

Once you've checked that it is indeed possible to take y and d close together, then that's that!

 

[stukbv]

arghh I am confused how would i check - sorry to be such a pain~!

 

[micromass]

Use the continuity of the inverse sine function...