Continuity
Continuity is a local property. A function \(f\) is continuous at an interior point \(c\) of its domain if
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
and left continuous thereby having a right endpoint \(c\) if
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 \(\forall x \in [a,b]\),
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, \([m,M]\).