Continuity of a discrete function

[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..??

 

[Hootenanny]

given a function F(x) = 1 ,x=1
2 ,x=2
3 ,x=3The above function is a 3 pointed graph. it is continuous . Is it just because every point has a specific value..please someone explain this..??

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

 

[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.''

 

[trambolin]

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

 

[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.

 

[n!kofeyn]

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.

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.

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.''

This is true for the simple cases, but not always. For example see the http://en.wikipedia.org/wiki/Cantor_function" . Also, the function defined is not piecewise continuous, but is continuous as mentioned above.

 

[peeyush_ali]

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.

u are right.. civilized

 

[peeyush_ali]

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

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