Prove $f(x) = 1/x$ is continuous at $x = 1/2$

Question

I need to write an $\epsilon, \delta$ proof. I know that $\delta$ must be less than $\frac12$, but I can't figure out the other $\delta$ in terms of $\epsilon$.

Thank you in advance!

Answer 1

$f(\frac{1}{2}) = 2 \to |f(x) - f(\frac{1}{2})| = \left|\dfrac{1}{x} - 2\right| = 2\dfrac{\left|\frac{1}{2}-x\right|}{|x|}$.

Let $\epsilon > 0$ be known, choose $\delta = \text{min}(\frac{1}{4},\frac{\epsilon}{8}) \to \left|x-\frac{1}{2}\right| < \delta \to |f(x) - f(\frac{1}{2})| < 2\cdot \dfrac{\delta}{\frac{1}{4}} = 8\delta < \epsilon$