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