Functions are not continuous in their work

Consider the transition from even $\lfloor x \rfloor$ to odd $\lfloor x \rfloor$ that is from 0 to 1 and from 2 to 3.

$\begin{cases} \lim_{\{x\} \to 1^-} \cos \left(\dfrac \pi 4(1-\{x\})\right) = \cos 0 = 1 \\ \sin \left(\dfrac \pi 4 (0) \right) = 0 \color{#D61F06} \ne 1 \end{cases} \implies$ no continuity.

Consider the transition from odd $\lfloor x \rfloor$ to even $\lfloor x \rfloor$ that is from 1 to 2.

$\begin{cases} \lim_{\{x\} \to 1^-} \sin \left(\dfrac \pi 4 \{x\} \right) = \sin \dfrac \pi 4 \color{#3D99F6} = \dfrac 1{\sqrt 2} \\ \cos \left(\dfrac \pi 4(1-0)\right) = \cos \dfrac \pi 4 \color{#3D99F6} = \dfrac 1{\sqrt 2} \end{cases} \implies$ continuity at $x=2$ .

Therefore, there is $\boxed 1$ point of continuity when $x$ is an integer.