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?
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instead you could use the other characterization of continuity. $\displaystyle \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.
Notice $\displaystyle |\frac{1}{x}-\frac{1}{x_0}|= \frac{|x-x_0|}{xx_0}< \frac{\delta}{xx_0} <\frac{2\delta}{x_0^2} <\varepsilon$ provided $\displaystyle \delta < \min \{ \frac{x_0}{2} , \frac{\varepsilon x_0^2}{2} \} $
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