$f(x) = \frac{1}{x-1}$ is considered continuous. It's not defined at $x = 1$, so it has a disconnected domain, but it is continuous at all values where it is defined.

More specifically, while $f(x)=\frac{1}{x-1}$ is discontinuous at $x=1$, it is a continuous function because of what @r~~ said.

But, Desmos doesn't say that https://math.codidact.com/uploads/BFqgLvA5GoQQdXzrgF1MUiz7 Earlier, I had took a picture from internet randomly

The graph of Desmos is correct. But the function is still a continuous function. Desmos "doesn't say" it is a continuous function, but it "doesn't say" that it is not a continuous function, either.

To me it's looking like discontinuous function. Here's another sample

Left sided one is discontinuous either. Am I understanding discontinuous wrong?

deleted user Yes, you understand continuous wrong. Continuity does not mean that the graph is path connected. Continuity means that you get arbitrary small changes of $f(x)$ for sufficiently small changes of $x$ inside the domain. The following would be a discontinuous function: $$f(x) = \begin{cases} \frac{1}{x-1} & \text{if $x\ne 1$}\\ 0 & \text{if $x=1$} \end{cases}$$

deleted user A function can be a continuous function and yet still have a discontinuity at a point, as long as that point is not in its domain.