Group Actions: Discontinuity

Question

Given a group action $G\curvearrowright X$.

Then it need not be a continuous one: $l_g\notin\mathcal{C}(X)$

As an example I have in mind: $$k\in\mathbb{Z}:\quad l_k(x\in\mathbb{Z}):=x+k,\quad l_k(x\notin\mathbb{Z}):=x\quad(x\in\mathbb{R})$$

Are there standard examples though?

Reference: This is a start-up to: Group Actions: Orbit Space

Answer 1

For now, let me give an answer due to Mike Miller.

(I will delete this one if he himself will post one.)

Choose a discontinuous bijection: $$\varphi:\mathbb{R}\leftrightarrow\mathbb{R}:\quad\varphi\notin\mathcal{C}(\mathbb{R})$$ Construct a group action by: $$\mathbb{Z}\curvearrowright\mathbb{R}:\quad l_k(x):=\varphi^k(x)$$ Then the left translations are discontinuous.