Continuity of Non-Fundamental Functions: A Theorem?

[Werg22]

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?

 

[whodoo]

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.

 

[StatusX]

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.

 

[Werg22]

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?

 

[HallsofIvy]

The composition of continuous functions is continuous. I think that is what you want.

 

[Werg22]

Ok how about the second question I asked? This one seems a little more interesting...

 

[HallsofIvy]

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.

 

[Werg22]

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?