Continuity on an interval and operations

Continuity on an interval

A function f is said to be continuous on an interval I if it is continuous at every point in I (with right- or left-sided continuity at any closed endpoints). The usual functions (polynomials, the exponential, sine, cosine, and square root on their domains) are continuous over their entire domain: these are results to be memorised rather than proved anew each time.

Operational theorems

If f and g are continuous at a, then:

• f + g and f * g are continuous at a
• f / g is continuous at a if g(a) ≠ 0
• g o f (the composite function) is continuous at a if f is continuous at a and g is continuous at f(a)

Example

h(x) = sqrt(x^2 + 1) is continuous throughout the entire real line: x → x^2 + 1 is continuous (polynomial) and strictly positive, sqrt is continuous on [0, +∞], so the composite function is continuous on the real line.

Summary table

Function	Domain of continuity
polynomial	ℝ
1/x	ℝ excluding 0
√x	[0, +∞[
ln(x)	]0, +∞[

Common pitfall

Be careful with the quotient: f/g is continuous only where g is non-zero. The function tan(x) = sin(x)/cos(x) is continuous on ℝ excluding the points where cos(x) = 0, but not on the entire ℝ. Always check the domain of definition before assuming the continuity of a quotient or a composite function.