Injective continuous function [closed]

I recently came across this question.

Let $\gamma : [0,1) \longrightarrow \mathbb{R}^k, k \geq 2$ be a injective and continuous function such that $Im(\gamma) \cap \omega(\gamma) = \emptyset$, where:

$$\omega(\gamma) = \{x \in \mathbb{R}^k : x = \lim \gamma(t_j), \text{for some sequence} [0,1) \ni t_j \rightarrow 1 \}.$$

How can I prove that $Im(\gamma) \cup \omega(\gamma)$ is closed and connected?

I'd be grateful if you could give me a tip!