= Discontinuity of a function
= Discontinuous
{synonym}
A <function> is <discontinuous> at a point in its <domain of a function> when it fails the <epsilon-delta definition of continuity> there. Two basic failures are an incorrect assigned value and unequal <one-sided limits>. They demand different repairs.
For a <removable discontinuity>, consider
$$
r(x)=\begin{cases}x^2,&x\neq0,\\1,&x=0.\end{cases}
$$
The <limit> at zero is $0$, but $r(0)=1$. Replacing that one value by $0$ gives the <continuous> square <function>. The <sequence> $x_n=1/n$ makes the mismatch visible: $r(x_n)=1/n^2\to0\neq r(0)$. A <removable discontinuity> can also describe an expression with a missing value, in which case the question is whether a <continuous> extension to the missing input exists.
For a <jump discontinuity>, take $j(x)=0$ for $x<0$ and $j(x)=1$ for $x\geq0$. The left and right <one-sided limits> at zero are $0$ and $1$. No assignment to $j(0)$ can equal both, so changing only the point value cannot make the <function> <continuous> there. Contrast both examples with the corner of $|x|$: that <function> is <continuous> at zero even though its <derivative> is absent.
\Image[/continuous-corner-removable-hole-and-jump.png]
{title=A continuous corner, a removable point-value mismatch, and a jump discontinuity}
{description=The three original <function graphs> separate <continuity> from <differentiability>. Open markers show excluded graph points; filled markers show assigned values. The jump has no vertical connector, because intermediate heights are not values of that <function>.}
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