Counterexamples about function discontinuity.

Question

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a function with a point $\textbf{x}\in\mathbb{R}^n$ of discontinuity. Is it possible that the image $f(O_{x_i})$, the image of an open ball (containing $x$) under $f$ to be connected for all $O_{x_i}$?

Answer 1

Yes. Take$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}\sin\left(\frac1x\right)&\text{ if }x\neq0\\0&\text{ otherwise.}\end{cases}\end{array}$$Then $f$ is discontinuous at $0$, but the image of every open interval containing $0$ is $[-1,1]$, which is connected.

Answer 2

It is even possible that $f:\mathbb R\to \mathbb R$ maps every interval of positive length onto $\mathbb R.$