I'm a university student studying mathematics. I'm currently doing a real analysis course and I'm working ahead on my problem sheet and I've arrived at this piecewise function.
$$f(x)= \begin{cases} x \cos(\frac{1}{x}) & \text{if }x \neq 0 \\ 2 & \text{if }x = 0 \\ \end{cases} $$
The question asks to prove that the function is continuous over $\mathbb{R} \backslash\{0\}$. Obviously the function breaks as it states it does at $x=0$ but barring the intuitive observation, if anyone has the rigorous technique for determining continuity of functions that would be of great help as we have not touched on it in class.
for the function to be continuous the left hand limit(LHD) must be equal to right hand limit(RHD) at x=o and also equal to f(0).
here clearly LHD and RHD tend to 0 as x approaches 0. here the function is discontinuous
at x=0.
you just need to evaluate LHD and RHD and compare them with value of function at that point.