What I have tried
Consider two arbitrary points on the unit circle, $P = (\cos \theta_1, \sin \theta_1)$ and $Q = (\cos \theta_2, \sin \theta_2)$, where $\theta_1$ and $\theta_2$ are angles in radians.
We can define a function $f: [0, 1] \rightarrow \mathbb{R}^2$ as follows:
$$ f(t) = (\cos((1-t)\theta_1 + t\theta_2), \sin((1-t)\theta_1 + t\theta_2)) $$
Here, $t$ varies from $0$ to $1$. When $t = 0$, $f(0) = P$, and when $t = 1$, $f(1) = Q$. The trigonometric functions $\cos$ and $\sin$ are continuous, and the composition of continuous functions is also continuous. Therefore, $f$ is continuous. So, the function $f$ defines a continuous curve between $P$ and $Q$ on the unit circle.
Is this correct?
One of my concern is: do I need to assume that $\theta_1 < \theta_2$?