Definition of Continuous Function

Definition: Continuous Function

A function $f\left(x\right)$ is continuous at a point c in its domain if

$\underset{x\to c}{\lim } f\left(x\right)=f\left(c\right)$

Note:

–          This is the definition you will need to apply when you are asked to, “use the formal definition” to show a function is continuous at a given x-value .

–          In more everyday language, a function is continuous at a specific x-value = c , when the limit value at that x-value, $\underset{x\to c}{\lim } f\left(x\right)$, equals the function value (y-value), $f\left(c\right)$, at that same x-value.

–          I like to say, “Does the limit equal where I would put the dot on the graph at that x-value ?”

–          When looking at this definition in terms of a graph, the places where you would have to lift your pencil to continue drawing the graph (holes, vertical asymptotes, jumps, breaks in the graph) are the places where the function is NOT continuous. These are the places where $\underset{x\to c}{\lim } f\left(x\right)\ne f\left(c\right)$.

–          Most parts of a function are usually continuous.