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

[rsa58]

Homework Statement

how do you show that sin(1/x) is continuous on (0,1)? (i know it's also continuous on (0, infinite)).

Homework Equations

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 similar 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

is this answer correct? what about the endpoints?

 

[rsa58]

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

 

[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)

 

[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?

 

[HallsofIvy]

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

 

[mathboy]

If a function is continuous on a set, A, it is continuous on any subset of A.

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?

 

[mathboy]

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

 

[morphism]

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

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

 

[morphism]

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?

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

 

[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.

 

[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.

 

[mathboy]

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

 

[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.