Definition of Continuity

Putting together the three requirements noticed in 8.1.1, a function $f(x)$ is said to be continuous at a point $x=a$ if all three of the following conditions hold:

1. $f$ is defined at every point of some open interval containing $a$ (so $f(a)$ itself is defined);
2. $\displaystyle\lim_{x\to a} f(x)$ exists;
3. $\displaystyle\lim_{x\to a} f(x) = f(a)$. Condition (iii) can be restated in a single equivalent line: $f(x)$ is continuous at $x=a$ if it is defined in some neighbourhood of $a$ and

   $\lim_{h\to0}\big[f(a+h)-f(a)\big]=0.$

   This says exactly the same thing — as the step $h$ away from $a$ shrinks to zero, the change in the function's value must also shrink to zero, i.e. there is no sudden jump right at $a$. Illustration 1. Let $f(x)=|x|$, defined on all of $\mathbb{R}$ (Fig. 8.4), so that $f(x)=-x$ for $x<0$ and $f(x)=x$ for $x\ge0$. Then

   $\lim_{x\to0^-} f(x)=\lim_{x\to0}(-x)=0,\qquad \lim_{x\to0^+} f(x)=\lim_{x\to0}(x)=0,$

   so both one-sided limits agree and $\displaystyle\lim_{x\to0} f(x)=f(0)=0$. Hence $f(x)=|x|$ is continuous at $x=0$. …