Let $A$ a subset of $\mathbb{R}$, let $f\colon A\to\mathbb{R}$ and let $x_0\in\ A.$ We say that $f$ is continuous in $x_0$ if for all $\varepsilon>0$ exists $\delta >0$ such that
$$x\in A, |x-x_0|<\delta\Rightarrow |f(x)-f(x_0)|<\varepsilon$$
On same book I read that $$x\in A, |x-x_0|\le\delta\Rightarrow |f(x)-f(x_0)|\le\varepsilon$$
What is the difference? Thanks!
Assume the first statement is true and let $\epsilon>0$ be given. Then there exits $\delta'>0$ such that $$x\in A, |x-x_0|<\delta'\Rightarrow |f(x)-f(x_0)|<\epsilon.$$ Let $\delta=\dfrac{\delta'}{2}>0.$ Then $$x\in A, |x-x_0|\le\delta\implies |x-x_0|<\delta'\Rightarrow |f(x)-f(x_0)|\le\epsilon.$$
Now assume the second is true and let $\epsilon >0$ be given. Then by the second statement, corresponding to $\dfrac{\epsilon}{2}>0,$ there exits $\delta>0$ such that $$x\in A, |x-x_0|\le\delta\Rightarrow |f(x)-f(x_0)|\le\frac{\epsilon}{2}<\epsilon.$$
Hence both statements are equivalent.
