Continuity

Let \(X \subseteq \mathbb{R}\) be a subset of real numbers, \(x_0 \in X\) and \(f: X \rightarrow \mathbb{R}\) be a function that maps elements of \(X\) to real numbers. The function \(f\) is continuous at \(x_0\) if and only if the limit of \(f(x)\) as \(x \rightarrow x_0\) is equal to \(f(x_0).\)

\[\lim_{x \rightarrow x_0} f(x) = f(x_0)\]

This uses the limit of a function, which requires \(x_0\) to be a limit point of \(X.\) If \(x_0\) is an isolated point of \(X\) (some open interval around \(x_0\) contains no other point of \(X\)) then \(f\) is continuous at \(x_0\) by convention.

The function \(f\) is continuous on \(X\) if and only if \(f\) is continuous at \(x \, \, \forall \, x \in X.\) Otherwise, \(f\) is discontinuous.

Continuity of composition

Let \(X \subseteq \mathbb{R}\) be a subset of real numbers. \(X\) is an interval if for all \(x,y \in X\) with \(x<y,\) every \(z \in \mathbb{R}\) with \(x<z<y\) is also in \(X.\) The empty set is an interval since it has no such \(x,y.\)

\[[x,y]\subseteq X \,\, \forall \, x,y \in X\]

Boundedness of intervals

The intermediate value theorem states that if a function \(f:[a,b]\rightarrow\mathbb{R}\) is continuous and \(C\) lies strictly between \(f(a)\) and \(f(b)\) (either \(f(a)<C<f(b)\) or \(f(b)<C<f(a)\)) then there is at least one \(c\in(a,b) : f(c)=C.\)

Functions of intervals

Continuity of intervals

Uniform continuity

Lipschitz continuity

Continuity of monotonic functions

Continuity of inverse functions