A discontinuity occurs at \(x=a\) when a function is not continuous there. Continuity requires all three conditions:
- \(f(a)\) exists;
- \(\lim_{x\to a}f(x)\) exists;
- \(\lim_{x\to a}f(x)=f(a)\).
Different failures produce different types of discontinuity.
1. Removable discontinuity: a hole
Consider
\[f(x)=\frac{x^2-1}{x-1}.\]
For \(x\ne1\), the expression simplifies to \(x+1\), but the original function is undefined at \(x=1\). The graph is the line \(y=x+1\) with a hole at \((1,2)\).
The limit exists:
\[\lim_{x\to1}\frac{x^2-1}{x-1}=2,\]
but \(f(1)\) does not. Defining \(f(1)=2\) would remove the discontinuity.
2. Jump discontinuity
For
\[g(x)=\begin{cases}x+1,&x<0,\\x+3,&x\ge0,\end{cases}\]
the one-sided limits are
\[\lim_{x\to0^-}g(x)=1,\qquad \lim_{x\to0^+}g(x)=3.\]
Because the one-sided limits are finite but unequal, the two-sided limit does not exist. The graph jumps from one height to another.
3. Infinite discontinuity
For \(h(x)=1/(x-2)\), the function grows without bound near \(x=2\):
\[\lim_{x\to2^-}h(x)=-\infty,\qquad \lim_{x\to2^+}h(x)=+\infty.\]
The line \(x=2\) is a vertical asymptote, and the function has an infinite discontinuity there.
4. Oscillatory discontinuity
Not every non-removable discontinuity is a jump or vertical asymptote. The function
\[q(x)=\sin\left(\frac1x\right),\qquad x\ne0,\]
oscillates infinitely often as \(x\to0\). The limit does not exist because the values do not approach one number or one infinity.
5. A reliable classification procedure
- Check whether the formula is defined at \(x=a\).
- Compute or estimate the left-hand and right-hand limits separately.
- If they agree and are finite, compare the common limit with \(f(a)\).
- If they are finite but unequal, classify a jump.
- If either side is unbounded, classify an infinite discontinuity.
- If the values oscillate without settling, classify an oscillatory discontinuity.
6. Graphing cautions
A plot may connect points across a narrow discontinuity if the sampling interval is too coarse. Zoom near the suspected x-value and use analytic limits to confirm. A removable hole can be difficult to see unless you add an open-point marker. A vertical asymptote should be distinguished from a steep but finite segment.
Use the Graphing Calculator to compare the examples. For the step-by-step construction of branches, see piecewise functions; for vertical asymptotes, see the asymptote guide.