# Proof that a function is continuous on its domain

1. Apr 30, 2013

### Whistlekins

1. The problem statement, all variables and given/known data

We have $f(x) = \frac{x^{2}+x-2}{x-1}+cos(x) , x\in\mathbb{R}\setminus \{1\}$ and wish to prove that it is continuous on its domain.

2. Relevant equations

The delta-epsilon definition of the continuity of a function.

3. The attempt at a solution

I've managed to reduce $|f(x) - f(x_0)|$to$|x-x_0| + |cos(x) - cos(x_0)| < \delta + |cos(x) - cos(x_0)|$
I'm not too sure where to go from there or even if I'm on the right track. Any insight would be greatly appreciated.

2. May 1, 2013

### Dick

Factoring x^2+x-2 would be a great first step.

3. May 1, 2013

### Whistlekins

I have, that's how I arrived at the reduced expression. The questions is where to go from there. I could always use the property that if $f$ and $g$ are continuous at a point $x_0\in \mathbb{A}$ then $f+g$ is continuous at $x_0$. But I don't know how to prove that $cos(x)$ is continuous on the domain.

4. May 1, 2013

### Dick

Yes, that you did. One way to prove cos is continuous is to use the trig identity, cos(u)-cos(v)=(-2)sin((u+v)/2)*sin((u-v)/2).

5. May 1, 2013

### Whistlekins

Ahh I overlooked that, thanks! I think I've got it now.

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