# Math Help - Continuity

1. ## Continuity

Using the definition of continuity directly prove that f
: (0,infinity) -->R defined by

f
(x) := 1/x is continuous.

I know you have to start with epislon>0 and finding a delta, but I can't come up with the right delta. Help?

2. ## easier way

instead you could use the other characterization of continuity.
$\lim_{x\to c} f(x) = f(c)$
you kno for x not equal to zero 1/x is exist in R.
now use algebraic limit theorem about reciprocal limits.

3. Notice $|\frac{1}{x}-\frac{1}{x_0}|= \frac{|x-x_0|}{xx_0}< \frac{\delta}{xx_0} <\frac{2\delta}{x_0^2} <\varepsilon$ provided $\delta < \min \{ \frac{x_0}{2} , \frac{\varepsilon x_0^2}{2} \}$