What are the points of discontinuity and non-differentiability of $| \sin(\pi/x)|$?
I tried finding out the points of discontinuity for the function but couldn't understand why would a mod function by discontinuous at all... Plz help me out , also if there are points of non discontinuity please tell how to solve such modulus ques
A discontinuity would be where the function in question does not have a defined value, or those where the function "jumps".
In that light, what would be the places where $f(x) = |\sin(\pi / x)|$ cannot be evaluated, or where are the jumps? We know $\sin(x)$ and $|x|$ are defined for all real $x$ and have no jumps, so the only possibility would be where the argument of the function, $\pi/x$, is undefined. In that light, it should be clear as to what the discontinuity of the function is.
A graph of the function will definitely prove useful as well.
As for determining where $f(x)$ is not differentiable, it might be easiest to first graph $f$. It'll have a bunch of a "sharp" points, which, if you remember discussions on differentiability from Calculus I and such classes, will be a sign of not being differentiable there. Finding $f'(x)$ explicitly will also help determine where the function is not differentiable as well, but I think the graphing method is sufficient in this case unless you want a fully-rigorous approach.
