Continuity

Continuity is a local property. A function f is continuous at an interior point c of its domain if

\lim_{x \to c} f(x) = f(c).

If either \lim_{x \to c} f(x) fails to exist or it exists but is not equal to f(c), then f is discontinuous at c.

Right and left continuity

f is right continuous at c thereby having a left endpoint c of its domain if

\lim_{x \downarrow c} f(x) = f(c)

and left continuous thereby having a right endpoint c if

\lim_{x \uparrow c} f(x) = f(c).

Continuity on an interval

f is continuous on the interval I if and only if f is continuous in each point of I. In endpoints left/right continuity is sufficient.

f is called a continuous function if and only if f is continuous on its domain.

Discontinuity

A discontinuity is removable if and only if the limit exists otherwise the discontinuity is non-removable.

Combining continuous functions

If the functions f and g are both defined on an interval containing c and both are continuous at c, then the following functions are also continuous at c:

the sum f + g and the difference f - g;
the product f g;
the constant multiple k f, where k is any number;
the quotient \frac{f}{g}, provided g(c) \neq 0; and
the nth root (f(x))^{\frac{1}{n}}, provided f(c) > 0 if n is even.

This may be proved using the various limit rules.

The extreme value theorem

If f(x) is continuous on the closed, bounded interval [a,b], then there exists numbers p and q in [a,b] such that \forall x \in [a,b],

f(p) \leq f(x) \leq f(q).

Thus, f has the absolute minimum value m=f(p), taken on at the point p, and the absolute maximum value M=f(q), taken on at the point q. This follows from the consequence of the completeness property of the real numbers.

The intermediate value theorem

If f(x) is continuous on the interval [a,b] and if s is a number between f(a) and f(b), then there exists a number c in [a,b] such that f(c)=s. This follows from the consequence of the completeness property of the real numbers.

In particular, a continuous function defined on a closed interval takes on all values between its minimum value m and its maximum value M, so its range is also a closed interval, [m,M].

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