Continuity of Non-Fundamental Functions: A Theorem?

In summary, the sum and product of two continuous functions is always continuous, as well as the difference and negation of a function. Compositions of continuous functions are also continuous in every topological space. If a function is continuous, it is equivalent to saying there exists a limit at every x. The composition of continuous functions is also continuous. If the functions involved in a composition have a finite number of oscillations on the desired intervals, the resulting function will also have the same property. There is no special theorem about this, as it follows directly from the regular laws of limits.
  • #1
Werg22
1,431
1
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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  • #2
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.
 
  • #3
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.
 
  • #4
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?
 
  • #5
The composition of continuous functions is continuous. I think that is what you want.
 
  • #6
Ok how about the second question I asked? This one seems a little more interesting...
 
  • #7
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.
 
  • #8
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?
 

1. What is continuity of functions?

Continuity of functions refers to the property of a function where there are no abrupt changes or breaks in the graph of the function. This means that as the input values of the function approach a certain point, the output values also approach a specific value, and there are no sudden jumps or gaps in between.

2. How is continuity of functions different from differentiability?

While continuity of functions refers to the smoothness of a function's graph, differentiability refers to the existence of the derivative of a function at a point. A function can be continuous but not differentiable, but if it is differentiable, it is also continuous.

3. What are the three types of continuity?

The three types of continuity are point continuity, uniform continuity, and global continuity. Point continuity refers to the continuity of a function at a specific point, uniform continuity refers to the continuity of a function over an entire interval, and global continuity refers to the continuity of a function over its entire domain.

4. How can you determine if a function is continuous?

A function is continuous if it satisfies the following three criteria: 1) the function is defined at the point in question, 2) the limit of the function as it approaches the point exists, and 3) the limit is equal to the value of the function at the point. If any of these criteria are not met, the function is not continuous.

5. What are some real-life applications of continuity of functions?

Continuity of functions is used in various fields such as engineering, physics, economics, and computer science. Some examples include designing smooth and continuous curves for bridges and roads, analyzing the flow of fluids through pipes, modeling financial data, and creating smooth animations in computer graphics.

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