Core idea: continuity via limits
\[
f \text{ is continuous at } a \iff
\lim_{x\to a^-} f(x)=\lim_{x\to a^+} f(x)=f(a).
\]
If any part of that statement fails, there is a discontinuity at (or near) \(a\).
Three main types
1) Removable discontinuity (a “hole”)
The left and right limits match and are finite, but \(f(a)\) is either undefined or not equal to the limit.
\[
\lim_{x\to a} f(x)=L \text{ (finite)}, \quad f(a) \text{ missing or } f(a)\neq L.
\]
A common cause is a cancellable factor in a rational function, like \((x-a)\) appearing in both numerator and denominator.
The calculator can also show a continuous extension by defining \(f(a)=L\) (“Fill the hole”).
2) Jump discontinuity
Both one-sided limits exist and are finite, but they are different:
\[
\lim_{x\to a^-} f(x)=L_-,\quad \lim_{x\to a^+} f(x)=L_+,\quad L_- \neq L_+.
\]
Typical examples include piecewise definitions and step-like functions (e.g., floor/ceiling).
3) Infinite discontinuity (vertical asymptote)
At least one one-sided limit diverges:
\[
\lim_{x\to a^-} f(x)=\pm\infty \quad \text{or}\quad \lim_{x\to a^+} f(x)=\pm\infty.
\]
For rational functions, this usually happens where the denominator is zero and not canceled.
If the factor \((x-a)^m\) remains in the denominator, the integer \(m\) is the (pole) multiplicity.
Rational functions: why factoring matters
If
\[
f(x)=\frac{P(x)}{Q(x)},
\]
then \(f\) is undefined wherever \(Q(x)=0\). Two outcomes are common:
-
Removable hole: \(P\) and \(Q\) share a factor \((x-a)\). After cancellation, the simplified function has a finite value at \(a\),
which equals the limit of the original.
-
Vertical asymptote: \(Q(a)=0\) but no cancellation occurs. Then \(f(x)\) typically blows up near \(a\).
The calculator’s Rational mode tries to parse expanded polynomials and detect common factors numerically/symbolically (best with integer coefficients).
If your input uses \((x-1)(x+1)\) style products, expand first for the cleanest factor steps.
