1.12 Confirming Continuity over an Interval

Learning objectives

—LIM-2.B—Determine intervals over which a function is continuous• LIM-2.B Determine intervals over which a function is continuous.• LIM-2.B.1 A function is continuous on an interval if the function is continuous at each point in the interval.• LIM-2.B.2 Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.• Enduring understanding LIM-2: Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.

A function is continuous on an interval when it is continuous at every point in that interval. Standard function families are continuous at all points in their domains, so begin by finding the domain.

Function family	Continuity domain check
Polynomial or exponential	Continuous wherever the formula is defined; standard real forms have no breaks
Rational	Exclude zeros of the denominator
Logarithmic	Require the logarithm's argument to be positive
Power	Apply the real-domain restrictions of the exponent and base expression
Trigonometric	Exclude inputs where the chosen trigonometric expression is undefined

Find every domain restriction.
Place the excluded values in order on the number line.
Use those values to split the domain into connected intervals.
State that the function is continuous on each remaining interval because its component functions are continuous there.

For $f(x)=\dfrac{\ln(x-1)}{x-3}$ , the logarithm requires $x>1$ and the denominator requires $x\ne3$ . Therefore the domain, and hence the intervals of continuity, are $(1,3)$ and $(3,\infty)$ .

“Continuous on its domain” does not mean continuous for every real number. An excluded input is not silently included in an interval, and a domain split must be written as separate intervals rather than one interval spanning the missing point.

Next Topic

—1.13 Removing DiscontinuitiesRead