# Does there exist a continuous bijective function $\displaystyle f : \Bbb R \rightarrow \Bbb R − \{1\}$?

This is my first time posting on here :)

My question is this: Does there exist a continuous bijective function $f :\Bbb R → \Bbb R − \{1\}$? Explain yes or no...

My thoughts:

Let $\displaystyle f: A \rightarrow B$ be a function

Okay for there to be a bijective function, for every element in A, there must be a unique element mapping $A$ to $B$. Right? Well if that is the case, then the answer to the question is FALSE. Why? Because the cadinality of the domain of $A$ (which in this case is R) is GREATER than the cardinality of the co-domain which is ($\Bbb R-\{1\}$).

Because this is so, there must be some duplicates such that two different elements in $A$ map to the same element in $B$.

Does this make sense?

Unfortunately, I would be happy with this answer but another question pops up in my mind. I have read in my textbook that the cardinality of $\Bbb N$ is equal to the cardinality of $\mathbb Z$

But how? Since $\Bbb N$ represents all natural numbers ($0,1,2,3...$) and $\mathbb Z$ represents all the integers ($...-3,-2,-1,0,1,2,3...$). Clearly the cardinality of $\mathbb Z$ is greater than N but still the two have equal cardinality.

So tying that in with the question, my answer doesn't seem right anymore.

Any guidence or help would be greatly appreciated!!!

Thanks :)

• Our intuition that a set cannot have the same cardinality as one of its subsets is true for finite sets, but breaks down when working with infinite sets. Mar 25, 2013 at 6:05
• That's even more than that: being equipotent to a proper subset is the definition of infinite in the sense of Dedekind. Mar 25, 2013 at 16:04

The cardinalities of $$\Bbb R$$ and $$\Bbb R\setminus\{1\}$$ are actually the same, even though it may seem that they shouldn't be. It's clear that $$\Bbb R$$ is "at least as big" as $$\Bbb R\setminus\{1\}$$, since the former contains the latter. On the other hand, the map $$\Bbb R\to\Bbb R\setminus\{1\}$$ given by $$x\mapsto1+e^x$$ is a one-to-one function, so $$\Bbb R\setminus\{1\}$$ is "at least as big" as $$\Bbb R$$. Thus, by this theorem, the two sets are "the same size".

The kicker here is that a continuous real-valued function on $$\Bbb R$$ can only have one of the following types of sets as its range:

(a) A singleton $$\{y\}$$--this happens for constant functions.

(b) An interval (half-open, open, or closed).

(c) A ray (open or closed).

(d) The whole real line.

$$\Bbb R\setminus\{1\}$$ isn't any of these types of sets, so no continuous bijection $$\Bbb R\to\Bbb R\setminus\{1\}$$ exists.

Alternately, we could use the Intermediate Value Theorem, which can also be used to prove the "kicker" mentioned above.

Let $$f$$ be any continuous function $$\Bbb R\to\Bbb R\setminus\{1\}$$. If $$f$$ takes on at least one value greater than $$1$$ and at least one value less than $$1$$, then there exist distinct $$a,b$$ such that $$f(a)<1. But since $$f$$ is continuous on the interval $$[a,b]$$ or $$[b,a]$$ (whichever one makes sense), then by IVT, there would then be some $$c$$ strictly between $$a$$ and $$b$$ such that $$f(c)=1$$, which is impossible, since $$f:\Bbb R\to\Bbb R\setminus\{1\}$$. Thus, for any continuous function $$f:\Bbb R\to\Bbb R\setminus\{1\}$$ either $$f(x)>1$$ for all $$x$$ or $$f(x)<1$$ for all $$x$$, so $$f$$ is not a bijection $$\Bbb R\to\Bbb R\setminus\{1\}$$. Thus, no continuous bijection $$\Bbb R\to\Bbb R\setminus\{1\}$$ exists.

• Exactly, what I was looking for! Thanks everyone for your help :) Really appreciate it! The information on "connected-ness" was slightly out of my reach but the Intermediate Value Theorem example made a lot of sense. Once again, thank you everyone! Mar 25, 2013 at 20:47

No. Because $\mathbb R$ is connected but $\mathbb R-\{1\}$ is not connected.

You can't do this by cardinality arguments, since $\mathbb{R}$ and $\mathbb{R}\setminus\{1\}$ actually have the same cardinality. You need some topological argument.

The keyword is connectedness. The continuous image of a connected set like $\mathbb{R}$ must be connected, unlike $\mathbb{R}\setminus\{1\}$. So your function can't even be continuous and onto.

Even though it boils down to the same in $\mathbb{R}$, it is slightly easier to consider convexity. Recall a set $S$ is convex in an affine space if, for every $s_1,s_2$ in $S$, and every $t\in [0,1]$, the element $(1-t)s_1+ts_2$ belongs to $S$. That is, $[s_1,s_2]\subseteq S$ as soon as $s_1,s_2$ belong to $S$.

By the intermediate value theorem, the continuous image of a convex set in $\mathbb{R}$ is convex. Now $\mathbb{R}$ is convex, but $\mathbb{R}\setminus\{1\}$ is not. So there does not even exist a continuous function from $\mathbb{R}$ onto $\mathbb{R}\setminus\{1\}$.

The concept of cardinality does not consider any particular structure imposed on a set (though they may help us count). In other words, it is the extra structure (such as operations or order relations) that misled you to a wrong claim that $\Bbb{Z}$ has more element than $\Bbb{N}$. Only when you obliterate this extraneous information you can truly compare the cardinality of two sets.

For example, let us rename each integers as follows:

$$\Bbb{Z} = (0, 1, -1, 2, -2, 3, -3, \cdots) = (a_1, a_2, a_3, a_4, a_5, a_6, a_6, \cdots).$$

In this way we can forget everything about the size, order and arithmetic relationship of the elements of $\Bbb{Z}$. Then now we identify the function $n \mapsto a_n$ as a bijection between $\Bbb{Z}$ and $\Bbb{N}$, hence conclude that they have the same cardinality.