12.3 Continuity  (Page 5/10)

Determining whether a function is continuous

To determine whether a piecewise function    is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.

Key concepts

• A continuous function can be represented by a graph without holes or breaks.
• A function whose graph has holes is a discontinuous function.
• A function is continuous at a particular number if three conditions are met:
  • Condition 1: $f(a)$ exists.
  • Condition 2: $\underset{x\to a}{\lim }f(x)$ exists at $x=a.$
  • Condition 3: $\underset{x\to a}{\lim }f(x)=f(a).$
• A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
• A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See [link] .
• Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See [link] and [link] .
• For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See [link] and [link] .

Section exercises

Verbal

Algebraic

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.