1

Let me state the context.

A function satisfying *** is called continuous.

Then I have a question. If I want to talk about properties that all continuous functions have, then which of the following two statement should I use?

  1. Properties of a continuous function are ***
  2. Properties of continuous functions are ***
  • Since you said "all continuous functions" there must be more than one function. Unless the "properties" are all the same among functions, use: "Properties of continuous functions are ***". If not, use the other one. – user3169 Jul 29 '16 at 2:32
2

Let us suppose that there are three properties: humpiness, bumpiness, and lumpiness. Your two sentences would be formed as follows:

  1. (The) properties of a continuous function are humpiness, bumpiness, and lumpiness.
  2. (The) properties of continuous functions are humpiness, bumpiness, and lumpiness.

Either one is acceptable and correct, and their meanings are the same. Choosing one or the other is a matter of style and preference. Because "properties" is a plural noun, you can also choose to include or omit The at the beginning of the sentence, depending upon your preference.

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  • Is it strange if I mix these two styles in a document? Or is it considered good because of a variety of expressions? – Eng Jul 29 '16 at 3:08
  • @Eng - It is probably best to be consistent throughout a piece of writing. Choose a style and then stick with it. – P. E. Dant Reinstate Monica Jul 29 '16 at 3:16
  • One more question. Is it the same if I change "properties" and "are" to "The definition" and "is"? Both are OK? – Eng Jul 29 '16 at 22:42
  • @Eng - That change would be fine, as long as there is only one definition. You could not write: The definition of a continuous function is humpiness, bumpiness, and lumpiness. You can say: The definition of a continuous function is that it is humpy, bumpy, and lumpy. – P. E. Dant Reinstate Monica Jul 29 '16 at 22:47

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