# Continuity of a discrete function

1. Jul 11, 2009

### peeyush_ali

given a function F(x) = 1 ,x=1
2 ,x=2
3 ,x=3

The above function is a 3 pointed graph. it is continuous . Is it just because every point has a specific value..please someone explain this..??

2. Jul 11, 2009

### Hootenanny

Staff Emeritus
That function is not continuous, at least not on the reals (it is however, continuous on the integers in the range [1,3]).

Last edited: Jul 11, 2009
3. Jul 11, 2009

### Fenn

Generally, when you talk about a continuous function, you can imagine a graph of the function being drawn with a single curve. In your case, it jumps from integer values, so it would be discontinuous in real space.

I believe the term for the function you have is generally referred to as piecewise continuous.''

4. Jul 11, 2009

### trambolin

It might be pointwise equal to a continuous function but, since it is not stated, no conclusion can be drawn.

5. Jul 11, 2009

### Civilized

Any function with a discrete domain is continuous on its domain, since you have defined the function on a discrete domain, it is mathematically correct to say that the function is continuous everywhere in its domain.

6. Jul 11, 2009

### n!kofeyn

Civilized is right. On the domain {1,2,3}, the function is continuous. It isn't even defined on other domains, so you can't make a determination of its continuity on the other domains.
This is true for the simple cases, but not always. For example see the http://en.wikipedia.org/wiki/Cantor_function" [Broken]. Also, the function defined is not piecewise continuous, but is continuous as mentioned above.

Last edited by a moderator: May 4, 2017
7. Jul 13, 2009

### peeyush_ali

u are right.. civilized

8. Jul 13, 2009

### peeyush_ali

continuity exists in the function just because the function value is mapped in its domain (in this case "specifically...")