# Discontinuous composite of continuous functions

• M Dhanota
In summary, the conversation discusses finding an example of two continuous functions, f and g, at x=0, where the composite function f(g(x)) is discontinuous at x=0. The question also arises if this contradicts the sandwich theorem and the conversation touches on the idea that one of the functions may also be discontinuous at a different point.
M Dhanota

## Homework Statement

give an example of functions f and g, both continuous at x=0, for which the composite f(g(x)) is discontinuous at x=0. Does this contradict the sandwich theorem? Give reasons for your answer.

## The Attempt at a Solution

I understand the sandwich theorem, but I don't understand how I can just make up these functions!

If f(g(x)) is discontinuous, what can we say about f(x) or g(x)?

I really want to say that one of the functions, either f(x) or G(x) is also discontinuous at a certain point. But not at x=0.

## 1. What is a discontinuous composite of continuous functions?

A discontinuous composite of continuous functions is a mathematical concept where two or more continuous functions are combined to create a new function that is not continuous at one or more points. This means that there is a break or gap in the graph of the composite function, even though each individual function is continuous.

## 2. How is a discontinuous composite of continuous functions different from a continuous composite of discontinuous functions?

A discontinuous composite of continuous functions is different from a continuous composite of discontinuous functions because in the former, the individual functions are continuous but the composite function is not, while in the latter, the individual functions are discontinuous but the composite function is continuous.

## 3. What are some real-life examples of discontinuous composite of continuous functions?

One example is the function representing the distance traveled by a car over time, where the car changes speed at different intervals. Another example is the function representing the height of a person jumping on a trampoline, where the height changes abruptly due to the elasticity of the trampoline.

## 4. How is a discontinuous composite of continuous functions used in scientific research?

In scientific research, a discontinuous composite of continuous functions can be used to model and analyze complex systems where there are multiple factors affecting a particular phenomenon. By combining several continuous functions, researchers can gain a better understanding of how different variables interact with each other.

## 5. Can a discontinuous composite of continuous functions ever be continuous?

No, a discontinuous composite of continuous functions can never be continuous. This is because the composite function inherits the discontinuities of the individual functions, and adding more continuous functions cannot eliminate these discontinuities.

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