53 lines
2.3 KiB
Markdown
53 lines
2.3 KiB
Markdown
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# Continuity
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Continuity is a local property. A function $f$ is continuous at an interior point $c$ of its domain if
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$$\lim_{x \to c} f(x) = f(c).$$
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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 $c$.
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## Right and left continuity
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$f$ is **right continuous** at $c$ thereby having a left endpoint $c$ of its domain if
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$$\lim_{x \downarrow c} f(x) = f(c)$$
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and **left continuous** thereby having a right endpoint $c$ if
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$$\lim_{x \uparrow c} f(x) = f(c).$$
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## Continuity on an interval
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$f$ 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.
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$f$ is called a continuous function if and only if $f$ is continuous on its domain.
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## Discontinuity
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A discontinuity is removable if and only if the limit exists otherwise the discontinuity is non-removable.
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## Combining continuous functions
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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 $c$:
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* the sum $f + g$ and the difference $f - g$;
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* the product $f g$;
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* the constant multiple $k f$, where $k$ is any number;
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* the quotient $\frac{f}{g}$, provided $g(c) \neq 0$; and
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* the *n*th root $(f(x))^{\frac{1}{n}}$, provided $f(c) > 0$ if $n$ is even.
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This may be proved using the various [limit rules](limits.md/#limit-rules).
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## The extreme value theorem
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If $f(x)$ is continuous on the closed, bounded interval $[a,b]$, then there exists numbers $p$ and $q$ in $[a,b]$ such that $\forall x \in [a,b]$,
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$$f(p) \leq f(x) \leq f(q).$$
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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.
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## The intermediate value theorem
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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.
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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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