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Can you give me examples of two functions $f$ and $g$ such that both are non-zero continuous function but their product is zero.

Topology
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    A function is actually a representation of real numbers which follows a rule known as "relation". So, if the functions do not reach $0$, i.e., the functions are non-zero, then multiplying two non-zero real numbers to get a function which reaches zero at some point looks impossible to me. – Hawk Feb 15 '14 at 15:16
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    To avoid much of the discussion in the answers below, the poster should disambiguate the "non-zero" property of the functions. Whether it means non-zero functions (f,g are non-zero somewhere in their domain, but they can have roots), or functions that are nowhere zero (they cannot have any root). From usual parlance one would understand it is the former, for which the answers are correct. – JuanPi Oct 26 '22 at 09:59

3 Answers3

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Consider $$f(x)=\max\{x,0\},\quad g(x)=\min\{x,0\}.$$

Hagen von Eitzen
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Define:

$f(x)=0$ for $x<0$ and $f(x)=\sin{x}$ for $x\geq 0$,

$g(x)=\sin{x}$ for $x<0$ and $g(x)=0$ for $x\geq 0$.

Then $f$, $g$ are continuous, non-zero functions and $(fg)(x)=0$ for all $x\in\mathbb{R}$.

alans
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    How is this non-zero if I am not missing something obvious?For $x=\pi$ and $x=-\pi$ the functions $f$ and $g$ are zero respectively. – Hawk Feb 15 '14 at 15:06
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    @Hawk A function is "non-zero" if it has non-zero value for at least one value of the argument. – David Mitra Feb 15 '14 at 15:07
  • @DavidMitra A function is non-zero not for all $x$? – Hawk Feb 15 '14 at 15:08
  • @DavidMitra I looked up on google, it says non-zero function is not $0$ for all of its domain...so am I missing something you are saying to me? – Hawk Feb 15 '14 at 15:12
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    @Hawk Hmm... Indeed, the problem is ambiguous. The answer here interprets it as "neither of the functions are the zero function". Though I think this is the intended meaning, your interpretation is reasonable as well. – David Mitra Feb 15 '14 at 15:22
  • @DavidMitra Thank you for agreeing...I was getting a little confused about everyone thinking in the same way and I am the only one to think differently indicates that I was going wrong somewhere. – Hawk Feb 15 '14 at 15:25
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A function is actually a representation of real numbers which follows a rule known as "relation". So, if the functions do not reach $0$, i.e., the functions are non-zero, then multiplying two non-zero real numbers to get a function which reaches zero at some point looks impossible to me.

EDIT:

Unless there are some imaginary number involved, the range is supposed to be in real numbers.

Hawk
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  • If you look at the question it does not say at any given point? – Math137 Feb 15 '14 at 15:22
  • @math137 non-zero functions refers to a function that does not reach zero at any point. – Hawk Feb 15 '14 at 15:23
  • can you give me a reference for that? – Math137 Feb 15 '14 at 15:25
  • @math137 http://wiki.answers.com/Q/What_is_a_nonzero_function – Hawk Feb 15 '14 at 15:27
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    @Hawk: I'd avoid using sites like 'Wiki Answers' as your authority on mathematical definitions. (For example, an answer on this page uses the other definition of 'nonzero'.) The 'zero function' is the function which takes the value $0$ everywhere; the natural interpretation of 'nonzero function' is a function which takes a nonzero value somewhere. – Clive Newstead Feb 15 '14 at 15:31
  • @CliveNewstead I actually did not want to use the website but I read it in a book, from which I got the information, and since I cannot quote it here...so I opted for 'Wiki Answers' to prove my point – Hawk Feb 15 '14 at 15:33
  • @Hawk: Which book? This use of 'nonzero' is pretty standard. – Clive Newstead Feb 15 '14 at 17:04
  • @CliveNewstead I read this in my basic 12th standard book, Calculus by Maron, and maybe some other books which I do not remember now. – Hawk Feb 16 '14 at 04:33