Definitions

Definitions are similar to theorems, but are true in both directions; technically, this means that the statement and its converse are both true ($p\leftrightarrow q$). The double arrow is read “if, and only if.” Both parts are either true or both parts are false. Definitions usually name some thing or some property.  Definitions are not proved.

The definition of continuity is a good example: A function f is continuous at xa if, and only if, these three things are true

(1)  $f\left( a \right)$ exist (i.e. is a finite number)

(2)  $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exist (i.e. is a finite number)

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

“Play” with it: consider cases where only 2 of the 3 requirements are true – is the function still continuous? What would happen if you removed the requirements about finite numbers?

To use a theorem one must be sure all the hypotheses are true. To use a definition one may say that either part is true once you have established that the other part is true. So if you know a function is continuous at a point, then the three statements are true; or if you can show the  three statements are true, you may say the function is continuous.

Here’s an example: A typical AP problem might give a piecewise defined function and ask if it is continuous at the place where the domain is divided.

To get credit for justifying an answer of “yes”, students must show that all the requirements of the definition are met. Specifically, they must show that the limit as x approaches that point must equal the value of  the function at that point (and both are finite).  In turn, to show that this limit exist the student must show that the hypotheses of the theorem that says if the two one-sided limits are equal to the same number, then that number is the limit.

To get credit for an answer of “no”, the student must show that (only) one of the hypotheses is false.

Finally, as with theorems, express definitions in words. With your students, “play” with the theorem or definition by making changes to the hypotheses and seeing how that affects the conclusion. Look at graphs. Don’t just state the definition and expect students to understand it, remember it and use it correctly.