Removable Discontinuity

Some discontinuities are much milder than a jump: the function's limit at the point does exist, but either the function was never given a value there, or it was assigned a value different from that limit. In either case, simply defining or redefining $f(a)$ to equal the limit patches the function into a continuous one. This is called a removable discontinuity:

In such a case, defining or redefining $f(a)$ as $\displaystyle\lim_{x\to a} f(x)$ makes the new function continuous at $x=a$; this repaired function is called the removable discontinuity, and if the original function was not defined at $a$ at all, the new, patched definition is called the extension of the original function.

Illustration 6. Consider $f(x)=\dfrac{x^2+3x-10}{x^3-8}$, for $x\ne2$; here $f(2)$ is not defined, since the denominator vanishes at $x=2$. Factorising, $x^2+3x-10=(x-2)(x+5)$ and $x^3-8=(x-2)(x^2+2x+4)$, so for $x\ne2$,

$\lim_{x\to2} f(x)=\lim_{x\to2}\frac{(x-2)(x+5)}{(x-2)(x^2+2x+4)}=\lim_{x\to2}\frac{x+5}{x^2+2x+4}=\frac{2+5}{4+4+4}=\frac{7}{12}.$ …