Continuity

A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function." A continuous function with a continuous inverse function is called "bicontinuous." Continuity of functions is one of the core concepts of topology.

Example: Consider the function $h(t)$, which describes the height of a growing flower at time $t$. This function is continuous. In fact, a dictum of classical physics states that in nature everything is continuous. By contrast, if $M(t)$ denotes the amount of money in a bank account at time $t$, then the function jumps whenever money is deposited or withdrawn, so the function $M(t)$ is discontinuous.

The function $f$ is continuous at some point $c$ of its domain if the limit of $f(x)$ as $x$ approaches $c$ through the domain of $f$ exists and is equal to $f(c)$. In mathematical notation, this is written as $\lim_{x \to c}{f(x)} = f(c)$.

In detail this means three conditions: 

1. $f$ has to be defined at $c$,
2. the limit on the left-hand side of that equation has to exist, and
3. the value of this limit must equal $f(c)$.

The function $f$ is said to be continuous if it is continuous at every point of its domain. If the point $c$ in the domain of $f$ is not a limit point of the domain, then this condition is vacuously true, since $x$ cannot approach $c$ through values not equal to $c$.