# Continuity of sin(1/x) on (0,1)

1. Dec 20, 2007

### rsa58

1. The problem statement, all variables and given/known data
how do you show that sin(1/x) is continuous on (0,1)? (i know it's also continous on (0, infinite)).

2. Relevant equations

3. The attempt at a solution

|f(x)-f(xo)| = |sin(1/x)- sin(1/xo)|= |2sin((xo-x)\2)cos((xo+x)/2)|

=< 2|sin((xo-x)/(2xox))|=< |(xo-x)/(xox)|. is this inequality true? a similiar one is used in a different example. if it is, why? is it because sin(x)=<x ? when x is positive? now since x<1 choosing $$\delta$$=xo$$\epsilon$$ then if

|xo-x|<$$\delta$$ then |f(xo)-f(x)|<$$\epsilon$$

2. Dec 20, 2007

### rsa58

i guess this is true for all xo and x different from 0 so it is true for (0, infinite) right?

3. Dec 20, 2007

### chickendude

A cheap way would be to use the theorem that differentiability implies continuity
The derivative of $$\sin(\frac{1}{x})$$ is $$\frac{-1}{x^2}\cos(\frac{1}{x})$$.
The only place where the derivative is undefined is at x=0 which is not in the given interval so the function is continuous on the interval.

I don't really like that way because it seems out of order (differentiability comes after continuity)

Here's another.
Definition of Continuity: The function is defined at every point on the interval and
$$\lim_{x\to c}f(x) = f(c)$$

We know that the function is defined at every point except 0, so it is defined in the interval.
The limit as x goes to c of the function is $$\sin(\frac{1}{c})$$ which is defined and equal to f(c) at every point except c=0, which isn't in the interval
Therefore, the function is continuous on (0,1)

4. Dec 20, 2007

### mathboy

sinx is continuous, and 1/x is continuous on (0,1). The composition of continuous functions is continuous. So sin(1/x) is continouous on (0,1)

Are you required to prove that sinx itself is continuous?

5. Dec 20, 2007

### HallsofIvy

Staff Emeritus
You say "(i know it's also continous on (0, infinite))". If a function is continuous on a set, A, it is continuous on any subset of A.

6. Dec 20, 2007

### mathboy

ANY subset of A? What if the subset has an empty interior? How can f inverse of an open subset of f(A) (in the subspace topology of R) be open in A?

Last edited: Dec 20, 2007
7. Dec 20, 2007

### mathboy

What's the epsilon-delta proof for the continuity of sinx itself?

8. Dec 20, 2007

### morphism

One can prove that |sin(x) - sin(y)| <= |x - y|.

9. Dec 20, 2007

### morphism

The empty set and A itself are still open subsets of A.

10. Dec 20, 2007

### mathboy

Ok, I got it. If A is given the subspace topology of R, then any continuous function restricted to A is still continuous.

But an epsilon-delta proof of this fact will not work if A is a set with an empty interior.

Last edited: Dec 20, 2007
11. Dec 25, 2007

### rsa58

i see you have all posted great replies, but can anyone tell me if my proof actually works. i want to understand the proper use of the epsilon-delta proof. the teacher will surely ask us to do such a proof. it seems okay.

12. Dec 25, 2007

### mathboy

There's no need to use the epsilon-delta proof, and I don't think your prof expects it.

13. Dec 25, 2007

### jacobrhcp

I don't think so either, the exercise does not seem to encourage you to take a look at the epsilon-delta kind of proofs.