If $A \subset R^{n}$ is not closed, show that there is a continuous function $f: A \rightarrow R$ which is unbounded.

Question

Question: 1-28. If $A \subset R^{n}$ is not closed, show that there is a continuous function $f: A \rightarrow R$ which is unbounded.

Hint : If $x \in R^{n} - A$ but $x \notin int(R^{n} - A)$, let $f(y) = \frac{1}{ \mid y-x \mid}$.

What I got so far:

Given $f(y) = \frac{1}{\mid y-x \mid}$, choose $p$ to be the point that's in the not closed set.

So, for $x \in A$ and $p \notin A$, our function becomes $f(x) = \frac{1}{\mid x-p \mid}$

By the continuous definition, for $ \epsilon > 0$ there exists a $\delta >0$ such that $\mid x-y \mid < \delta$ implies $\mid f(x)-f(y) \mid < \epsilon$.

$\mid \frac{1}{\mid x-p \mid} - \frac{1}{\mid y-p \mid} \mid < \epsilon$  

$\mid \frac{\mid y-p \mid}{\mid y-p \mid} \cdot \frac{1}{\mid x-p \mid} - \frac{1}{\mid y-p \mid} \cdot \frac{\mid x-p \mid}{\mid x-p \mid} \mid < \epsilon$

$\mid \frac{\mid y-p \mid -\mid x-p \mid }{\mid y-p \mid \mid x-p \mid } \mid < \epsilon$

$\mid \frac{\mid y-p-x+p \mid }{\mid y-p \mid \mid x-p \mid } \mid < \epsilon$

$\mid \frac{\mid y-x \mid }{\mid y-p \mid \mid x-p \mid } \mid < \epsilon$

I need to find two Deltas but I'm not sure how to do that. Does it have something to do with using

$\mid \mid x \mid - \mid y \mid \leq \mid x-y \mid$ ?

Or maybe I'm doing this backwards because if I take the negative out of the numerator, I'll end up with $-\mid -y+x \mid$.

Answer 1

Since $X \subseteq \mathbb{R}^n$ is not closed, there exists $a \in \mathbb{R}^n$ and a sequence $\{x_n\}$ in $X$ such that $\lim_{n\to\infty}x_n=a$ but $a \not\in X$. Without loss of generality, you can assume that $x_n$ is never equal to $a$ for any $n \in \mathbb{N}$.

Now consider $f(x) = \frac{1}{\|a-x\|}$. Since $\lim_{n\to\infty}x_n=a$, we know that:

$$\forall K \in \mathbb{N}, \exists M\in\mathbb{N}: n\geq M \implies \|a-x_n\|<\frac{1}{K}$$

This means that $$\forall K\in \mathbb{N}:|f(x_n)|=\frac{1}{\|a-x_n\|}>K$$

Hence, $f$ is not bounded.

To see that $f(x)$ is continuous in a punctured neighborhood of $a$, note that $f(x)$ is a rational function where the denominator is zero only at $x=a$. So, $f(x)$ is indeed continuous over $\mathbb{R}^n - \{a\}$. To follow an $\epsilon-\delta$ approach, you need to review and combine the following two proofs:

1. $f(x) = \frac{1}{x}$ is continuous over $\mathbb{R}-\{0\}$
2. If $g$ and $f$ are continuous at $x=c$, so is $f\circ g$.

Proving the second statement is straightforward and easy. I leave it to you. In our case, it suffices to show that $f(x) = \frac{1}{x}$ is continuous over $(0,\infty)$. We first observe that for a fixed value $y \in (0,+\infty)$, we have:

$$|f(x)-f(y)| = |\frac{1}{x}-\frac{1}{y}|=|\frac{y-x}{xy}|\leq\frac{|x-y|}{y}\cdot\frac{1}{x}$$ Now, our goal is to select $\delta$ such that it's easy to find an upper bound for $\frac{1}{x}$ when $x \in (y-\delta,y+\delta)$. Equivalently, we want to choose $\delta$ such that $x \in (y-\delta,y+\delta)$ is away from $0$. Choosing $\delta < \frac{y}{2}$ should do the job for us. Indeed, you can easily show that in this case we have $\frac{1}{x} < \frac{2}{y}$. This means that

$$|f(x)-f(y)| \leq \frac{|x-y|}{y}\cdot\frac{2}{y} \leq \frac{2\delta}{y^2}$$

Now choosing $\delta < \min\{\frac{y}{2},\frac{y^2}{2} \epsilon\}$ finishes the proof. Q.E.D.

Answer 2

Supongamos que $A\subset\mathbf{R}^{n}$ no es cerrado. Entonces, existe un punto $x\in \mathbf{R}^{n}\setminus A$ que pertenece a la adherencia de $A$. Esto significa que para cualquier vecindad $V$ de $x$, existe algún punto $y\in V\cap A$.

Definamos ahora la función $f:A\to \mathbf{R}$ como $f(y) = \frac{1}{\lvert y - x\rvert}$. Notemos que esta función está bien definida ya que $x\notin A$.

Por otro lado, veamos que $f$ es continua. Sea $y_{0}\in A$ y $\epsilon > 0$. Debemos encontrar $\delta > 0$ tal que $\lvert y - y_{0}\rvert < \delta$ implica que $\lvert f(y) - f(y_{0})\rvert < \epsilon$.

Para ello, tomamos $\delta = \frac{1}{\epsilon}$. Entonces, si $\lvert y - y_{0}\rvert < \delta$, se tiene que:

$$\lvert f(y) - f(y_{0})\rvert = \left\lvert\frac{1}{\lvert y - x\rvert} - \frac{1}{\lvert y_{0} - x\rvert}\right\rvert \leq \frac{\lvert \lvert y - x\rvert - \lvert y_{0} - x\rvert\rvert}{\lvert y - x\rvert \lvert y_{0} - x\rvert}$$

Pero como $y_{0}\in A$, se tiene que $\lvert y_{0} - x\rvert > 0$, y por lo tanto:

$$\frac{\lvert \lvert y - x\rvert - \lvert y_{0} - x\rvert\rvert}{\lvert y - x\rvert \lvert y_{0} - x\rvert} \leq \frac{\lvert y - y_{0}\rvert}{\lvert y - x\rvert \lvert y_{0} - x\rvert} < \frac{\delta}{\delta^{2}} = \epsilon$$

Por lo tanto, $f$ es continua en $A$. Sin embargo, notemos que $f$ es no acotada ya que para cualquier $y\in A$, se tiene que $\lvert f(y)\rvert > \lvert f(x)\rvert = \frac{1}{\lvert x - y\rvert}$. Como $x$ pertenece a la adherencia de $A$, existen puntos de $A$ arbitrariamente cercanos a $x$, y por lo tanto $f$ no está acotada en $A$.