Continuous Functions (College Board AP® Calculus AB)

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Proving that a given function is continuous on all points of its domain can become quite tedious
- Luckily, there are standard results that you can use for many commonly-occurring types of function

A polynomial function is a function made up of sums or differences of positive integer powers of x, along possibly with constant terms
- These are polynomial functions:
  - $f (x) = x^{2} - 6 x + 15$
  - $g (x) = x^{5} - x^{4} + 3 x^{2} + x$
- These are not polynomial functions:
  - $h (x) = x^{3} - \frac{1}{x}$
    - Because $\frac{1}{x} = x^{- 1}$ is not a positive integer power of $x$
  - $j (x) = x^{2} + \sqrt{x} - 7$
    - Because $\sqrt{x} = x^{\frac{1}{2}}$ is not a positive integer power of $x$
Polynomial functions are continuous on all points in their domains
- This includes the domain of all real numbers
- Or any smaller domain defined for a particular function

A rational function is a function defined as a fraction (or quotient), where the expressions in the numerator and denominator are both polynomials
- E.g. $\frac{x^{4} + 2 x^{2} - 3}{x - 7}$ is a rational function
A rational function is continuous everywhere, except at points where its denominator is equal to zero
- E.g. $\frac{x^{2} + 7}{x^{2} - x - 2} = \frac{x^{2} + 7}{(x + 1) (x - 2)}$ is not continuous at $x = - 1$ or $x = 2$
- However it is continuous over all other real numbers
- Or any smaller domain not containing $x = - 1$ or $x = 2$
The graphs of rational functions have vertical asymptotes at points where their denominators are equal to zero

An exponential function is a function of the form $k e^{p x}$ or $k a^{p x}$ , where $k$ and $p$ are non-zero real number constants, $a$ is a positive real number constant, and $e$ is the exponential constant $e = 2.7182 . . .$
- I.e. a function where the variable $x$ appears as an exponent (or power)
Exponential functions are continuous on all points in their domains
- This includes the domain of all real numbers
- Or any smaller domain defined for a particular function
The graphs of exponential functions of the forms given above have horizontal asymptotes at $y = 0$

A logarithmic function is a function of the form $k \ln (p x)$ or $k \log_{a} (p x)$ , where $k$ and $p$ are non-zero real number constants, and $a$ is a positive real number constant
- I.e. a function where the variable $x$ appears inside a logarithm
- Remember that $\ln x = \log_{e} x$
Logarithmic functions of the above forms are continuous at all points where $p x > 0$
- E.g. $\ln (2 x)$ is continuous for all $x > 0$ ; $\ln (- 3 x)$ is continuous for all $x < 0$
- A smaller domain on the 'correct side of zero' may also be defined for a particular function
The graphs of logarithmic functions of the forms given above have vertical asymptotes at $x = 0$

Sine or cosine functions of the form $k \sin (p x)$ or $k \cos (p x)$ , where $k$ and $p$ are non-zero real number constants, are continuous on all points in their domains
- This includes the domain of all real numbers
- Or any smaller domain defined for a particular function
A tangent function of the form $k \tan (p x)$ , where $k$ and $p$ are non-zero real number constants, is continuous at all points where $p x$ is not an odd multiple of $\frac{π}{2}$
- I.e. at all points where $p x \neq . . ., - \frac{5 π}{2}, - \frac{3 π}{2}, - \frac{π}{2}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, . . .$
- At points where $p x = . . ., - \frac{5 π}{2}, - \frac{3 π}{2}, - \frac{π}{2}, \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, . . .$ , a tangent function of the above form has a vertical asymptote

the (exam) results speak for themselves:

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