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Let $f: X \rightarrow Y$ be such that for every connected subset $C$, $f (C)$ is connected. Can we conclude that $f$ is continuous? I think this is false, at the time of building a counterexample in $\mathbb{R}$ or $\mathbb{R}^2$ it gives me that it's true so I'm not sure anymore.

Kevin Alonso
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$X=Y=\Bbb R$ and $f$ being your favourite discontinuous derivative of some function will do.

  • Nice. For if $g:\Bbb R\to \Bbb R$ is differentiable then $g$' has the IVP (Intermediate Value Property) but $g'$ need not be continuous. A common example is $g(0)=0$ and $g(x)=x^2 \sin 1/x$ when $x\ne 0.$ – DanielWainfleet May 19 '19 at 16:18