5

I'm au courant with https://math.stackexchange.com/a/288133 and https://math.stackexchange.com/a/422001. They're both Abbott P111 exm 4.3.6 which proves "a continuous function is sometimes described, intuitively, as one whose graph can be drawn $\color{seagreen}{by \; dint \; of}$ without lifting your pencil from the paper. I countenance this.

But I don't think the underneath is a counterexample? I see $f(n) = n$ for all natural numbers $n$ is discontinuous over $\mathbb{R} \times \mathbb{R}$. But what if I visualize over $\mathbb{N} \times \mathbb{N}$? Then $f(n) = n$ can be drawn without lifting your pencil ?

enter image description here

(p 1 sur 14 https://math.la.asu.edu/~dajones/class/371/ch4.pdf)

Community
  • 1

2 Answers2

7

The intuitive idea that continuity means "able to be drawn without lifting your pencil" only makes sense for functions from $\mathbb R$ to $\mathbb R$ since the graph of these functions is a curve on a plane and, for geometric purposes, a plane is the same as $\mathbb R\times\mathbb R$. For other types of functions, the idea doesn't make sense. The graph of a function from $\mathbb N$ to $\mathbb N$, for instance, consists of discrete points, and there is no way to draw any such graph without lifting your pencil since your pencil is not allowed to touch the space between the lattice points (points with integer coordinates): that empty space is not properly part of $\mathbb N\times\mathbb N$. As another example, consider a real function of two real variables, that is, a function from $\mathbb R\times\mathbb R$ to $\mathbb R$. The graph of that function is a surface, but of course you cannot draw a surface with a pencil! The fact that our pencil analogy breaks down for these types of functions explains why we need a better definition of continuity, the modern definition: even if you could make the pencil idea mathematically precise, it can't directly be applied to most functions in general.

user134824
  • 12,212
5

I believe it was Euler who, in the 18th century, described a continuous function as one which could be drawn by "freely leading the hand". This is the "no pencil lifting" description.

This "definition" turned out not to be precise enough, and in the 19th century it was replaced with the $\epsilon$-$\delta$ definition. That it turn was generalized in the 20th century, when metric spaces and topological spaces were introduced.

As it happens, the old intuitive "no pencil lifting" notion doesn't match the precise definitions perfectly. There are pretty dramatic examples showing the mismatch: there are continuous functions that are differentiable nowhere, and there is a continuous function from the unit interval onto the (filled in) unit square, $f:[0,1]\rightarrow [0,1]\times[0,1]$.

Notice that Abbott says "is sometimes described, intuitively, as..." You can't prove that an imprecise intuitive notion is mathematically equivalent to a formal precise definition. So the words "proved" and "theorem" are inappropriate in this context.

Intuition, however, can be retrained and refined. Here's a way to think about the continuity of the function $f:\mathbb{N}\rightarrow\mathbb{N}$, $f(n)=n$. It's the restriction to $\mathbb{N}$ of the function $g(x)=x$, $g:\mathbb{R}\rightarrow\mathbb{R}$. Obviously $g$ is continuous, so it would be very surprising if $f$ turned out to be discontinuous.

In the context of general topology, the set $\mathbb{N}$ inherits a topology from $\mathbb{R}$; this inherited topology is the discrete topology on $\mathbb{N}$. Every function whose domain is a discrete topological space is continuous.

Michael Weiss
  • 8,813