Piecewise function - rational and irrational

In summary: As for part c), the limit is zero if and only if $x^2 = 0$. This is because $\lim_{x \to 0} x^2 = 0$, and $0$ is the only value of $x$ for which this is true.
  • #1
Dethrone
717
0
$$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$

a) Prove that $\lim_{{x}\to{0}}g(x)=0$
b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$

I've never seen a piecewise function defined that way...hints?
 
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  • #2
Hint: Use the $\delta-\varepsilon$ definition of limit.

Let $\varepsilon > 0$ be given, choose $\delta = [...]$. Consider the cases of $\left| f(x) - f(0) \right|$ if $x$ is rational and if $x$ is irrational.
 
  • #3
Darn! (Angry) I was planning to skip all the $\delta-\varepsilon$ questions because I haven't learned it yet, but seeing as you put in the time to answer my question and that I've already asked it, I will learn it first thing tomorrow morning (Nod)
 
  • #4
Here's a restate of the $\epsilon-\delta$ proof in a little nonrigorous language, see if you can verify it step-by-step :

Let $g_0$ be the function $g$ restricted to the set of rationals $\Bbb Q$. As $g_0(x)$ is just $x^2$ and $\lim_{x \to 0} x^2 = 0$, the limit point of $\{g_0(x_n)\}$ is $0$ for any sequence of rationals $\{x_n\}$ converging to $0$.

On the other hand let $g_1$ be the zero-map $x \mapsto 0$ restricted to the set of irrationals $\Bbb R - \Bbb Q$. As $\lim_{x \to 0} 0 = 0$, the limit point of $\{f(y_n)\}$ is $0$ for any sequence of irrationals $\{y_n\}$ converging to $0$.

Thus, as $g$ is the aggregate of the two functions $g_0$ and $g_1$, $\{g(z_n)\}$ has limit point $0$ for any sequence $\{z_n\}$ converging to $0$. A proper continuous choice of $\{z_n\}$ verifies that $\lim_{x \to 0} g(x)$ does indeed converge to $0$.
 
  • #5
Rido12 said:
$$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$

a) Prove that $\lim_{{x}\to{0}}g(x)=0$
b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$

I've never seen a piecewise function defined that way...hints?

Hi Rido12,

For part a), note that $0\le g(x) \le x^2$ for all $x\in \Bbb R$. So by the squeeze theorem, $lim_{x\to 0} g(x) = 0$.

For part b), you can argue by contradiction, using the fact that every interval contains an irrational number. Suppose $\lim_{x \to 0} g(x)$ exists with limit $L$. Then for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x$, $|x - 1| < \delta$ implies $|g(x) - L| < \epsilon$. Choose $\epsilon < 1/2$. If $L = 0$, then taking $x = 1$ forces $|g(x) - L| = 1 >\epsilon$. So $L$ is nonzero. Take an irrational number $x$ in $(1 - \delta, 1 + \delta)$. Then $|g(x) - L| < \epsilon$, whhich implies $|L| < \epsilon$. On the other hand, $|g(1) - L| < \epsilon$, that is, $|1 - L| < \epsilon$. Hence $\epsilon > |L| > 1 - \epsilon$, which yields $\epsilon > 1/2$ $(\rightarrow \leftarrow)$.
 

FAQ: Piecewise function - rational and irrational

What is a piecewise function?

A piecewise function is a mathematical function that is defined by different equations over different intervals of its domain.

What is a rational function?

A rational function is a function that can be written as the ratio of two polynomial functions. This means that the numerator and denominator of the function are both polynomial expressions.

What is an irrational function?

An irrational function is a function that cannot be written as the ratio of two polynomial functions. This means that the numerator and/or denominator of the function contain irrational numbers, such as pi or square root of 2.

How can I determine if a function is piecewise?

A function is considered piecewise if it is defined by different equations over different intervals. To determine if a function is piecewise, you can look for breaks or jumps in the graph of the function.

What are some applications of piecewise functions?

Piecewise functions are commonly used in mathematical modeling to represent real-world situations where different equations are applicable over different intervals. They are also used in computer programming to create more complex and efficient algorithms.

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