- #1

- 726

- 1

## Main Question or Discussion Point

Say f(x) = x^2 - 1 and I'm trying to prove that f is continuous, then I was told I CANNOT do this:

[tex] |x^2 - x_0^2| = |x-x_0||x+x_0| < \delta|x+x_0| = \epsilon [/tex]

because then our epsilon is relying on an x value. I was told I could restrict the x values to a closed neighborhood about the point x = x_0, and let epsilon rely on the end points, but I cannot let it rely on any x values like in the example shown.

Is this correct? And if it is, why is it?

[tex] |x^2 - x_0^2| = |x-x_0||x+x_0| < \delta|x+x_0| = \epsilon [/tex]

because then our epsilon is relying on an x value. I was told I could restrict the x values to a closed neighborhood about the point x = x_0, and let epsilon rely on the end points, but I cannot let it rely on any x values like in the example shown.

Is this correct? And if it is, why is it?