Can Two Discontinuous Functions Become Continuous When Added Together?

In summary, the conversation discusses finding two discontinuous functions that, when added together, become continuous. One approach is to let f be a discontinuous function and g be 1-f. It is then proven that g is discontinuous and a continuous function can be found as the sum of two continuous ones. There is also a mention of defining functions and the importance of understanding what it means to do so.
  • #1
semidevil
157
2
just a basic question, so if I'm asked to find 2 functions that are discontinus, but when added together, becomes continuous, how do I approach that?

can I say like, let

F(x) = 1 for x =< 0, and f(x) = 0 for x > 1.
G(x) = 1 for x =<0 and g(x) = 0 for x = 1.

can I just somehow "add" f + g and say that is continuous? I don't know...tips?
 
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  • #2
Let f be your favourite discontinuous function, then let g=1-f.

Prove g is discontinuous, and hence find a continuous function that is the sum of two continuous ones.

Your f and g, did you mean to capitalize them? Note g doesn't have the same domain as f; g isn't defined for any positive real numbers, and hence neither is f+g.
 
  • #3
semidevil said:
just a basic question, so if I'm asked to find 2 functions that are discontinus, but when added together, becomes continuous, how do I approach that?

can I say like, let

F(x) = 1 for x =< 0, and f(x) = 0 for x > 1.
G(x) = 1 for x =<0 and g(x) = 0 for x = 1.

can I just somehow "add" f + g and say that is continuous? I don't know...tips?

Well, you can't just say it's continuous- because it isn't!

(f+ g(x)= 2 for x<0, 1 if x= 0 and 0 if x>0
(I've switched the last "1" to "0". The f and g you give are not defined between 0 and 1. Unless that's a typo, I have the uncomfortable feeling that you don't know what is meant by "defining" a function.

Taking f(x)= 1 for x<= 0, f(x)= 0 for x> 0, which is not continuous at x= 0,
try matt grimes' suggestion. What is g(x)= 1- f(x)?
 

What is discontinuity to continuity?

Discontinuity to continuity refers to a change or transition from a state of being disconnected or separate to a state of being connected or continuous.

What are some examples of discontinuity to continuity?

Examples of discontinuity to continuity can include a phase change in matter (such as the solid to liquid transition), a transformation of a function from being discontinuous to continuous, or a social or political movement that leads to greater unity and cohesion.

What causes a discontinuity to continuity?

A discontinuity to continuity can be caused by various factors, such as external forces (such as pressure or temperature changes), internal processes (such as chemical reactions or social movements), or gradual changes over time.

What is the significance of studying discontinuity to continuity?

Studying discontinuity to continuity can provide insights into the processes and mechanisms that drive changes and transitions in various systems and phenomena. It can also help us understand how disruptions and discontinuities can lead to new forms of continuity and integration.

What are some potential challenges in studying discontinuity to continuity?

Some challenges in studying discontinuity to continuity include identifying and defining the boundaries of a system or phenomenon, understanding the complex interactions and feedback loops involved, and accounting for the role of chance and unpredictability in the process of change.

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