Continuity of a discrete function

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Discussion Overview

The discussion revolves around the continuity of a discrete function defined at specific integer points. Participants explore the implications of defining continuity in the context of discrete versus continuous domains, and whether the function can be considered continuous based on its defined values.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that the function is continuous because it has specific values at each defined point.
  • Another participant counters that the function is not continuous on the reals, but may be continuous on the integers within the specified range.
  • A participant describes the general expectation of continuous functions being represented by a single curve, noting that the function in question jumps between integer values.
  • Some participants argue that any function with a discrete domain is continuous on that domain, suggesting that the function is continuous everywhere within its defined points.
  • There is a mention of the term "piecewise continuous" in relation to the function, though its applicability is debated.
  • One participant references the Cantor function as an example that complicates the notion of continuity in discrete cases.

Areas of Agreement / Disagreement

Participants express differing views on the continuity of the function, with some agreeing that it is continuous on its discrete domain while others maintain that it is not continuous when considering the real numbers. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

The discussion highlights the limitations of defining continuity based on the domain of the function, with some participants emphasizing the need for clarity on the context in which continuity is being evaluated.

peeyush_ali
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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..??
 
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peeyush_ali said:
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]).
 
Last edited:
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.''
 
It might be pointwise equal to a continuous function but, since it is not stated, no conclusion can be drawn.
 
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 said:
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.
Fenn said:
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.
 
Last edited by a moderator:
Civilized said:
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
 
Hootenanny said:
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...")
 

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