Continuity

Continuity turns out to be probably the most important property that a function can have. It says that to a large part, the function must behave like polynomials do with respect to limits. That is, we say that a function $f:I\rightarrow(-\infty,\infty)$, where I is an interval, is continuous at $a\in I$ if

$$\lim_{x\rightarrow a} f(x) = f(a).$$

That is, a function is continuous at a if its limit at a is the same as its value at a.

There is an important connection between continuity and rates of change.

Suppose that $f:I\rightarrow(-\infty,\infty)$ is given to us, I is an interval and $a\in I$. Consider the difference quotient

$$DQ(h):=\frac{f(a+h)-f(a)}{h}$$

Ignoring for a moment what exactly the domain might be, suppose we wanted to define a function g with rule

$$g(h) = \left\{ \begin{array} {cc} {\displaystyle \frac{f(a+h... ...h}} & {\rm if}\;h\neq 0\\ m & {\rm if}\;h = 0\end{array}\right.$$

Finding the value for m so that g is continuous at is the same as finding the rate of change of f. This is very important, and you should invest some time and effort into convincing yourself that it is true!