Can a nowhere continuous function have a connected graph?

After noticing that function f:RR f(x)={sin1xfor x00for x=0
has a graph that is a connected set, despite the function not being continuous at x=0, I started wondering, doest there exist a function f:XY that is nowhere continuous, but still has a connected graph?

I would like to consider three cases

  • X and Y being general topological spaces
  • X and Y being Hausdorff spaces
  • ADDED: X=Y=R

But if you have answer for other, more specific cases, they may be interesting too.

Answer

Here is an example for R2R:

f(x,y)={ywhen x=0 or x=1xwhen x(0,1) and y=01xwhen x(0,1) and y=x(1x)x(1x)when x{0,1} and y/x(1x)Q0otherwise

This is easily seen to be everywhere discontinuous. But its graph is path-connected.


A similar but simpler construction, also R2R:

g(1+rcosθ,rsinθ)=rfor r>0,θQ[0,π]g(rcosθ,rsinθ)=rfor r>0,θQ[π,2π]g(x,y)=0everywhere else

Attribution
Source : Link , Question Author : Adam Latosiński , Answer Author : hmakholm left over Monica

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