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Continuity

Tags

Calculus

Cegep/1

Word count

186 words

Reading time

2 minutes

Quality of a function:

• At a point $c$ if $\underset{x\to c}{lim}f(x)=f(c)$
• Over an open interval if every point of the interval is continuous
• Over a closed interval if the open interval is continuous and the left / right endpoints are right / left continuous

Sidedness

Left continuity at $c$ if $\underset{x\to c^{-}}{lim}f(x)=f(c)$
Right continuity at $c$ if $\underset{x\to c^{+}}{lim}f(x)=f(c)$

Properties & theorems

1. $f@g$ is continuous at $c$, where $@$ is any of addition, subtraction, multiplication or division (when $@$ = division, $g(c)\ne 0$), if $f$ and $g$ are continuous at $c$
2. $f\circ g$ is continuous at $c$ if $g$ is continuous at $c$ and $f$ is continuous at $g(c)$
3. Functions continuous over their domain:
   • Polynomials
   • Rational functions
   • Root functions
   • Trigonometric functions
   • Inverse trigonometric functions
   • Exponential functions
   • Logarithmic functions

Evaluation

To find if $f(x)$ is continuous at $c$, evaluate $\underset{x\to c}{lim}f(x)=f(c)$.

Contributors

Ajitani Hifumi

Changelog

Last edited about 2 hours ago

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