math: add limit continuity

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eggy 2020-10-05 11:35:18 -04:00
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@ -352,7 +352,7 @@ There may only be one-sided limits. In this case, breaking the limit up into its
#### Change in variable
Substituting a variable in for the variable to be solved and then solving in terms of that variable may help.
Substituting a variable in for the variable to be solved and then solving in terms of that variable may remove a problem variable.
??? example
$$
@ -367,6 +367,28 @@ Substituting a variable in for the variable to be solved and then solving in ter
= \frac{1}{12}
$$
### Limits and continuity
If a function has holes or gaps or jumps (i.e., if it cannot be drawn with a writing utensil held down all the time), it is **discontinuous**. Otherwise, it is a **continuous** function. A function discontinuous at $x=a$ is "discontinuous at $a$", where $a$ is the "point of discontinuity".
A **removable discontinuity** occurs when there is a hole in a function. It can be expressed as when either
$$
f(a) = \text{DNE or} \\
\lim_{x \to a} f(x) ≠ f(a)
$$
A **jump discontinuity** occurs when both one-sided limits have different values. It is common in piecewise functions. It can be expressed as when
$$\lim_{x \to a^-} f(x) ≠ \lim_{x \to a^+} f(x)$$
An **infinite discontinuity** occurs when both one-sided limits are infinite. It is common when functions have vertical asymptotes. It can be expressed as when
$$\lim_{x \to a} f(x) = ± ∞$$
Therefore, a function is only continuous if all of the following are true:
- $f(a)$ exists
- $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$
- $\lim_{x \to a} f(x) = f(a)$
## Resources
- [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)