Example of Discontinuous Function: f(x) & f(f(x))

In summary, a discontinuous function is a function with one or more breaks or jumps in its graph. An example is the step function, and the difference between it and a continuous function is the presence of breaks in the graph. For a discontinuous function, f(x) & f(f(x)) is the composition of the function with itself, which may result in multiple breaks in the graph. These functions can be identified by looking at their graphs or by using the intermediate value theorem.
  • #1
nietzsche
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Homework Statement



Give an example of a function f(x) that is discontinuous at each point of its domain, whereas f(f(x)) is continuous everywhere.

Homework Equations





The Attempt at a Solution



I guess the function has to be defined piecewise, but I don't know how to approach this at all. I'm thinking of making conditions for x rational, x irrational, but I wouldn't know how to make the function discontinuous at ALL irrational points. Maybe the function doesn't need to take on values at all x? I really don't know.
 
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  • #2


never mind, i figured it out
 

FAQ: Example of Discontinuous Function: f(x) & f(f(x))

1. What is a discontinuous function?

A discontinuous function is a function that has one or more breaks or jumps in its graph. This means that there are points in the domain of the function where the value of the function is undefined or changes abruptly.

2. What is an example of a discontinuous function?

An example of a discontinuous function is the step function, f(x) = {1 if x ≥ 0, 0 if x < 0}. This function has a jump at x = 0, where the value changes from 0 to 1.

3. What is the difference between a discontinuous function and a continuous function?

A continuous function is a function that has a smooth and unbroken graph, meaning there are no jumps or breaks in the graph. On the other hand, a discontinuous function has one or more points where the graph is not smooth or breaks.

4. What is f(x) & f(f(x)) for a discontinuous function?

f(x) & f(f(x)) is the composition of a function with itself. For a discontinuous function, this means plugging in the function into itself, which can result in multiple jumps or breaks in the graph.

5. How can we identify a discontinuous function?

A discontinuous function can be identified by looking at its graph and observing any jumps or breaks. Additionally, a function can be checked for continuity using the intermediate value theorem, which states that a continuous function will take on all values between any two points in its domain.

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