Continuity of Non-Fundamental Functions: A Theorem?

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Discussion Overview

The discussion revolves around the continuity of non-fundamental functions derived from fundamental functions through various operations such as addition, multiplication, and composition. Participants explore whether the continuity of these non-fundamental functions can be guaranteed under certain conditions and inquire about relevant theorems that support these claims.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions if non-fundamental functions are continuous if all fundamental functions involved are continuous on specified intervals, seeking theorems that support this.
  • Another participant suggests that the continuity follows from the laws of limits, indicating that the sum of two continuous functions is continuous, but questions if a specific theorem exists for this case.
  • A different participant clarifies that the sum and product of continuous functions are continuous, and that the composition of continuous functions is also continuous in any topological space.
  • One participant proposes a refined question regarding the continuity of functions that can be decomposed into monotonic intervals, asking if such functions retain monotonicity when composed.
  • There is a reiteration that the composition of continuous functions is continuous, but the discussion shifts to the implications of oscillation and monotonicity in the context of continuity.

Areas of Agreement / Disagreement

Participants generally agree on the continuity of sums, products, and compositions of continuous functions. However, there is no consensus on the implications of oscillation and monotonicity for the continuity of more complex functions, as well as the existence of specific theorems to support these ideas.

Contextual Notes

The discussion includes assumptions about the definitions of fundamental and non-fundamental functions, as well as the conditions under which continuity is preserved. There are unresolved questions regarding the specific theorems that may apply to the continuity of functions with finite oscillations.

Werg22
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For non-fundamental functions obtained by a set of fundamental functions (either by multiplication, addition, division, compound or all together), and given those fundamental functions are all continuous on the desired intervals, will those non-fundamental functions also be continuous? I know this is true for simple compound functions, but does it hold for every other transformation listed? If so, what are the names of the theorems that prove it?
 
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Doesnt it follow directly from the regular laws of limits? For instance if the limits f(x), x->a exists, and g(x), x->a exists, then the limit f(x) + g(x), x->a exists(and is of course equal to f(a)+g(a)). Perhaps I misunderstood the question.

I don't think there is a special theorem about this.
 
What do you mean by fundamental functions? In any case, the sum and product of two continuous functions is always continuous. So is the negation of a function, which entails that the difference of continuous functions is continuous. The reciprical of a function is continuous iff the function is never zero, and this gives you f(x)/g(x) is continuous when f(x) and g(x) are if g is never 0. Compositions of continuous functions are continuous in every topological space.
 
Ok, I just realized that the fact that a saying that a function is continuous is equivalent to saying there exist a limit at every X makes it so that resulting functions are also continuous. A better question would be, if a we have fundamental functions that are continuous on the desired intervals and can be decomposed into a sequence of monotonic intervals (meaning they have a finite number of oscillations), will the resulting function also be monotonic on selected intervals?
 
The composition of continuous functions is continuous. I think that is what you want.
 
Ok how about the second question I asked? This one seems a little more interesting...
 
Was the second question "If so, what are the names of the theorems that prove it?"

I just said- The composition of continuous functions is continuous.
 
No sorry if I wasn't very clear. To restate the second question: say we have a function that is composed of other continuous functions. If those functions have a finite number of oscillation on the concerned intervals (meaning if we subdivide those intervals properly, the functions will be monotonic on the subdivisions), will the the function in question also be the same?
 

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