This is part of a series on common misconceptions.
Is this true or false?
All continuous functions can be drawn without lifting the pen from the paper.
Why some people say its true: Because of the name "continuous," it seems to them that these functions can be drawn continuously without any interruption or break. Often students are taught in that way in their junior classes.
Why some people say its false: Well, people who have a very basic knowledge of calculus don't actually say this is false, since the idea that continuous functions are continuous in the aforesaid loose sense is sufficient for them to solve ordinary application-oriented sums.
The statement is . There are continuous functions which cannot be drawn without lifting the pen from the paper.
Let us first begin with the formal definition of a continuous function. Let and . Then is said to be continuous at if given an there exists a such that if and , then . If is not continuous at , then it is said to be discontinuous at . Now note that the definition does not make any mention of whether is a limit point of or not (limit points or cluster points are just points such that any arbitrarily small neighborhood of them contains infinitely many points in ). If is so, then we have to check the following three conditions to testify the continuity of at :
must be defined at , so that is meaningful.
The limit of at must exist in .
The limit value must equal the functional value at .
However, if is not a limit point of , we see that there exists a neighborhood of such that . Thus we conclude that is by default continuous at those points in its domain , which are not limit points of , i.e., which are "isolated points" of .
Let be a function such that . Then seems to be discontinuous but it is actually not. See that its domain consists of isolated points only.