Does it makes sense for a function to have a discrete range even though the range is continuous? If yes how is it defined, and is it called something specific?

To explain what I mean if one had to model time against whether the light is on or off (to indicate when light goes on and light goes off). The range will just be 0 and 1, nothing in between, while the domain is a continuous value, time.

  • $\begingroup$ You just gave an example. A related one is the floor function $\lfloor x\rfloor$, the greatest integer $\le x$. $\endgroup$ – André Nicolas Nov 21 '13 at 1:53
  • $\begingroup$ Yes, what I mean is, does it have a specific name? Is it defined in a specific way? It is not a continuous function, so I don't think it is defined using limits, or is it? $\endgroup$ – jbx Nov 21 '13 at 1:55
  • $\begingroup$ While a function can respect the continuity of its domain, it doesn't have to. Just like it doesn't have to respect order, or inequality, or all sorts of other things. I recommend checking out how a function is set-theoretically defined. $\endgroup$ – Malice Vidrine Nov 21 '13 at 2:03
  • $\begingroup$ There is no special name. Except in the trivial case, it cannot be continuous, but saying "not continuous" is definitely not enough. $\endgroup$ – André Nicolas Nov 21 '13 at 2:34

It makes sense and your example is a good one. A function is required to return a single value for each element of the domain, but doesn't have to be continuous. A couple other example of functions on $\Bbb R$ are $\lfloor x \rfloor$ and the function that is $1$ if $x$ is rational and $0$ otherwise.

  • $\begingroup$ So in my case, am I correct to say that it is not a continuous function? And the definition of it can still be: $lights: \Bbb R_{\geq0} \rightarrow \{0,1\}$ $\endgroup$ – jbx Nov 21 '13 at 2:00
  • 1
    $\begingroup$ Yes, that is correct. We seem to confuse people when we "draw a function " and always make it continuous, even differentiable. Most functions are not $\endgroup$ – Ross Millikan Nov 21 '13 at 2:06

I would say it has to do largely with whether the function is continuous. If a function on $I \subseteq \mathbb{R}$ is continuous, with $I$ an interval, its range is either a single value (discrete) or an interval.

This is because continuity preserves a property called "connectedness".

  • $\begingroup$ Yes, its those kinds of issues I am asking about. In the example I gave, if the function was $lights(t)$, $t$ is definitely a continuous value, but at the same time the result is just 0 or 1. So is it a continuous function or not? Or does it have a specific name? And how is it defined (do you put $lim$ etc.)? $\endgroup$ – jbx Nov 21 '13 at 1:57
  • $\begingroup$ $lights(t)$ is not continuous if the light ever changes state. You have defined it fine-you have explained what the value is for any $t$ $\endgroup$ – Ross Millikan Nov 21 '13 at 2:02
  • $\begingroup$ @jbx As Ross said, the function is discontinuous if the light switch ever changes state. $\endgroup$ – Eric Auld Nov 21 '13 at 2:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.