Continuity 2

The definition of continuity of a function used in most first-year calculus textbooks reads something like this:

A function f is continuous at x = a if, and only if,

(1) f(a) exists (the value is a finite number),

(2) \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) exists (the limit is a finite number), and

(3) \underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right) (the limit equals the value).

A function is continuous on an interval if, and only if, it is continuous at all values of the interval. For the endpoints of closed intervals, the limits are adjusted to one-sided limits with x approaching a from inside the interval.

As I’ve written before, limits logically come before continuity since limits are used in the definition of continuity. But as a practical and historical matter continuity comes first. Continuity, or rather lack of continuity, gives us the examples that motivate the need for the concept of limit.

Karl Weierstrass (1815 – 1897) gave the modern definition of continuity: Given a function f and an element a of the domain I,   f is said to be continuous at the point a if for any number \varepsilon >0, however small, there exists a number \delta >0 such that for all x in the domain of f\left| x-a \right|<\delta  implies \left| f\left( x \right)-f\left( a \right) \right|<\varepsilon .

This looks very much like the definition of limit. In the delta-epsilon definition of limit the last inequality above is \left| f\left( x \right)-L \right|<\varepsilon  where L is the limit. Replacing the value with the limit allows a somewhat simpler wording of the definition of continuity than that given at the beginning, but adds the delta-epsilon complication. Weierstrass’ definition eliminates the need for saying the value and the limit are finite since that is assumed by writing f (a).

Some textbooks use the phrase “a function is continuous on its domain.” This seems somewhat limiting (no pun intended) in that a function such as \displaystyle f\left( x \right)=\tfrac{1}{x} is certainly continuous on its domain but not continuous on the entire number line. We are usually concerned about where a function is not continuous, so first we find where it is not continuous: at the points not in its domain plus possibly other points in its domain.

Recent AP calculus exams (2012 AB4c, 2011 AB 6a) gave students a piecewise defined function and asked if it is continuous at the point where the two pieces meet. The question directed students to “use the definition of continuity to explain your answer” and “show that f is continuous.”  To answer the question students were expected to state what the two one-sided limits are and what the value there is. Since all these numbers are finite and equal the requirements of the definition are met.

This kind of question could be considered a question about continuity or a question about applying a definition (or theorem) to a particular situation. Either way students should understand the hypotheses of a definition or theorem and know how to verify that they are met in a particular situation.


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