Connected graph of real function, but function not continuous

Question

I am reading Strichartz' The Way of Analysis. He mentions in the beginning of chapter on contionuous functions that

"Now it is indeed possible to make a precise mathematical definition of connected sets in the plane so that the graph of a continuous function is connected. However, it turns out that there are functions that are not continuous whose graphs are also connected. "

I cannot imagine any of such graphs. Can I get example of a function, preferably, from $\Bbb R$(or its subset) to $\Bbb R$ such that graph is connected while function is not continuous?

https://math.stackexchange.com/questions/1243983/graph-connected-does-not-imply-f-is-continuous — B.Swan, Aug 30 '17 at 15:00

score 2 · Accepted Answer · edited Sep 02 '17 at 14:39

2

I think that $f:\mathbb{R}\to\mathbb{R}$ defined by $f(0) = 0$ and $f(x) = \sin(\dfrac1x)$ for $x\neq 0$ should work. It's not continuous at $x = 0$ and its graph is connected, I think.

edited Sep 02 '17 at 14:39

Nosrati

30,522

answered Sep 02 '17 at 14:33

user476879

116

Connected graph of real function, but function not continuous

1 Answers1