Calculus I · Limits and Continuity · reference
Which Functions Are Continuous?
Use this Which Functions Are Continuous? reference to choose methods, review notation, and solve Calculus I limits and continuity problems.
Where this chapter fits
Chapter 5: Continuity
Connect limits to function values, classify discontinuities, repair piecewise definitions, and use the Intermediate Value Theorem.
Reading lens: Do the limit, the function value, and the surrounding domain fit together at the point or across the interval? Keep that question in view while reading Which Functions Are Continuous?; the worked mathematics is evidence for the idea, not a substitute for it.
This page connects Continuity on Intervals and at Endpoints to Types of Discontinuity. Read the explanation first, predict each example’s next move, and only then compare the written solution.
Families of continuous functions
The following functions are continuous everywhere in their natural domains:
• polynomials; • rational functions where the denominator is nonzero; • root functions where the root is real; • trigonometric functions wherever defined; • exponential functions; • logarithmic functions for positive arguments.
Sums, differences, products, constant multiples, and valid compositions of continuous functions are continuous. Quotients are continuous where their denominators are nonzero.
Find intervals of continuity
Find the intervals on which
is continuous.
Show worked solution
The square root requires
so
The denominator must be nonzero:
so
The value is already outside the square-root domain. The only excluded domain point within is .
Therefore, the function is continuous on
At , continuity is interpreted from the right.
Composition
Where is
continuous?
Show worked solution
The logarithm requires a positive argument:
This means
so
The polynomial is continuous everywhere, and is continuous for . Therefore the composition is continuous on
Source & rights
Original instruction with traceable references.
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