Continuety of sum of functions

In summary, the conversation is discussing whether the sum of two functions, f(x) and g(x), will continue to be continuous at a given point, X0, if one or both of the individual functions are not continuous at that point. The speaker suggests constructing a counter example to support the answer and then proving the result. The other person mentions considering functions that have discontinuities but still sum to a constant, but is unsure how to prove this. The first speaker doubts the need for proof and suggests using the given example to come up with a counterexample using the same logic.
  • #1
transgalactic
1,395
0
A.does f(x) +g(x) continues in X0 when
f(x) continues in X0 but g(x) doesnt?

B.does f(x) +g(x) continues in X0 when both
f(x) and g(x) are not continues in X0?

prove and give exmple to support your answer

??
 
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  • #2
If you're uncertain whether or not each function is continuous, try to constuct a counter example satisying the initial premises. After that, prove whatever result you obtained.
 
  • #3
For b) consider functions that have discont. but for example sum to be some constant.
 
Last edited:
  • #4
i don't know how to prove such things
??
 
  • #5
Are you sure you are supposed to prove them? b) is false using the example I gave so come up with a counterexample using the logic that I told you
 

FAQ: Continuety of sum of functions

What is the continuity of a sum of functions?

The continuity of a sum of functions refers to the smoothness and connectedness of the resulting function when two or more functions are added together. It is a property that indicates how well the function behaves and transitions between different points on its domain.

How is the continuity of a sum of functions determined?

The continuity of a sum of functions is determined by taking the limits of each individual function and then adding them together. If the resulting function has a limit at a point, then it is considered to be continuous at that point.

What happens when two continuous functions are added together?

If two continuous functions are added together, the resulting function will also be continuous. This is because the sum of two continuous functions will still have a limit at every point on their shared domain.

Can a sum of discontinuous functions be continuous?

Yes, it is possible for a sum of discontinuous functions to be continuous. This can occur when the discontinuities of each individual function cancel out when they are added together, resulting in a continuous function.

Is the continuity of a sum of functions affected by the order of addition?

No, the order in which the functions are added does not affect the continuity of the resulting function. As long as all of the individual functions are continuous, the sum will also be continuous regardless of the order in which they are added.

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