Definitions name things; mathematical definitions name things very precisely.
A good definition (in mathematics or anywhere) names the thing defined in a sentence that,
- Puts the thing into the nearest class of similar objects.
- Gives its distinguishing characteristic (not all its attributes, only those that set it apart).
- Uses simpler (previously defined) terms.
- Is reversible.
An example from geometry: A rectangle is a parallelogram with one right angle.
THING DEFINED: rectangle.
NEAREST CLASS OF SIMILAR OBJECTS: parallelogram.
DISTINGUISHING CHARACTERISTIC: one right angle. The other characteristics – the other three right angles, opposite sides parallel, opposite sides congruent, etc. – can all be proven as theorems based on the properties of a parallelogram and the one right angle. No need to mention them in the definition. This also helps keep the definition as short as possible.
PREVIOUSLY DEFINED TERMS: parallelogram, right angle. These you are assumed to know already; they have been previously defined.
IS REVERSIBLE: This means that, if someone gives you a rectangle, then without looking at it you know you have a parallelogram and it has a right angle, AND if someone gives you a parallelogram with a right angle, you may be absolutely sure it is a rectangle. In fact, you could write the definition the other way around: A parallelogram with one right angle, is a rectangle. Either way is okay.
Definitions often use the phrase … if, and only if.... For example: A parallelogram is a rectangle if, and only if, it has a right angle. The phrase indicates reversibility: the statement and its converse (and therefore, its contrapositive and inverse) are true.
When you get a new term or concept defined in calculus (or anywhere else), take a minute to learn it. Look for the nearest class of similar things, its distinguishing characteristics, and be sure you understand the previously defined terms. Try reversing it; say it the other way around. At that point you’ll pretty much have it memorized.
Definitions are never proved. There is nothing to prove; they just name something. Statements in mathematics that need to be proved are called theorems.
Finally, you may take the words in bold above as the definition of a definition!
Teaching Calculus 