Is the function $\sin^{x}(x)$ discontinuous on the entire $\mathbb{R}$? I have come across the function $\sin^{x}(x)$ (or a in a different notation$(\sin(x)^{x})$  while practicing calculating its the first derivative. I then plot it in Desmos since I don't know how to graph this by hands yet. It seems that the graphing device suggests this function is discontinuous between the interval $[\pi ,2\pi ]$. The graph is also peculiar between the interval $[0 ,\pi ]$, while replicating itself for any interval $(2\pi ,3\pi )$, $(3\pi ,4\pi )$, $(5\pi ,6\pi )$, etc.
The behaviour of this function on the  entire interval $(-\infty ,0)$ is also interesting. It doesn't look anywhere similar to  $[0,+∞)$.
The first derivative of this function is $\sin^{x}(x)+\ln(\sin(x))+\frac{x\cos(x)}{\sin(x)}$. As with most other complex transcendental functions, this is not a simple derivative to work with. Could you show me some strategies to explore and graph this function manually. I would like to particularly find the red circle point found on this graph:

I have also tried to plot this on Wolfram and its shape is similar. Is this an accurate shape at all or is the graphing device errorneous somehow?

Note that the function is not defined on reals for


*

*$\sin x=x=0$ (but $\sin^x x \to 1$ for $x\to o^+$)

*$\sin x<0$ (negative base is not well defined)

