Is $\csc(x)$ a continuous function?

Question

I have come across a past paper question in which asks to show there is a root between interval [1.2,1.3] in the function $f(x) = 4\csc(x) - 4x + 1$ using the change of sign method, however, I know this method only works for continuous functions. I'm pretty certain that the asymptotic nature of $\csc(x)$ would render this method invalid as it means that the function is discontinuous yet it still asks for it?

Answer 1

f(x) is continuous at points where $\csc x$ is continuous.

Note that $\csc x = \frac {1}{\sin x}$ is continuous at points where $\sin x \ne 0$

Since $$\sin x =0 \iff x=k\pi$$ $\csc x$ is continuous on $[1.2,1.3]$

Answer 2

Since $f$ is only discontinuous at the multiples of $\pi$, it i continuous on the interval of interest.