Given any function that maps connected sets to connected sets, is it continuous?

Question

Suppose $f: X \to Y$ is a function with the property that any connected subset of $X$ is mapped to a connected subset of $Y$. Does it follow that $f$ is continuous?

Answer 1

What about

$$f(x) = \begin{cases} \sin \left(\frac{1}{x}\right) & x \neq 0\\ 0 & x=0 \end{cases}$$

defined on $\mathbb R$. $f$ is not continuous at $0$. However if $I$ is a non empty interval, $f(I)$ is an interval for $0 \notin I$ as $f$ is continuous on $\mathbb R \setminus \{0\}$. And $f(I) = [-1,1]$ if $ 0 \in I$.