Working with Piecewise Functions

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In summary, we need to prove that the given Piecewise function is continuous only at the point $x=\frac{1}{2}$, using the fact that a function is continuous at a point if for every sequence converging to that point, the sequence of function values also converges to that point. To prove this, we can choose sequences of rational and irrational numbers converging to a given point and show that the corresponding sequences of function values converge to the same point. The fact that $f(x)=f^{-1}(x)$ is not helpful in this case.
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Enzipino
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I'm given the following Piecewise function when $f:[0,1]\to[0,1]$:
$f(x) = x$ when $x\in\Bbb{Q}$
$f(x) = 1-x$ when $x\notin\Bbb{Q}$

I need to prove that $f$ is continuous only at the point $x=\frac{1}{2}$.

For this problem, I know I need to use the fact that a function $f$ is continuous at a point $x$ iff for every sequence ${{x}_{n}}$ that converges to a number $x$ as $n\to\infty$ we also have the sequence $f({x}_{n})$ converging to $f(x)$ as $n\to\infty$. So, my initial approach to this was to first assume that $x\ne\frac{1}{2}$. And then I let $x\in\Bbb{Q}$ and I choose a sequence of irrational numbers ${x}_{n}$ that converges to $x$ as $n\to\infty$. But then from here I don't know how to work with the functions of the sequences. Do I have the right idea for this?

I also wanted to know that if I could simply just say that $f(x)= f^{-1}(x)$ once I prove that the function is bijective? (This is a separate problem but involves using the same piecewise function.
 
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  • #2
Hi,

I don't think the fact that $f(x)=f^{-1}(x)$ is helpful there, but it's correct.

For the first part, notice that given a point $x\in [0,1]$ you can choose a sequence of rational numbers $(y_{n})$ converging to $x$ and you will get a sequence $(f(y_{n}))$ converging to $x$ and you can also choose a sequence of irrational ones $(z_{n})$ and you will get a sequence $(f(z_{n}))$ converging to $1-x$.
 

FAQ: Working with Piecewise Functions

What is a piecewise function?

A piecewise function is a mathematical function that is defined by different rules or equations for different parts of its domain. This allows the function to have different behaviors or outputs for different input values.

How do you graph a piecewise function?

To graph a piecewise function, you can break it down into its individual parts and graph each part separately. Then, you can combine the individual graphs to get a complete graph of the piecewise function.

What is the purpose of using a piecewise function?

Piecewise functions are useful for representing real-life situations where different rules or conditions apply for different parts of the situation. They can also be used to represent discontinuous or non-linear functions.

What are some common applications of piecewise functions?

Piecewise functions are commonly used in fields like physics, engineering, economics, and computer science to model real-world phenomena that have different behaviors under different conditions. They are also used in data analysis to represent complex data relationships.

How do you find the domain and range of a piecewise function?

The domain and range of a piecewise function can be determined by looking at the domain and range of each individual part of the function. The domain is the set of all possible input values, while the range is the set of all possible output values.

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