Discontinuity

Sort by the limit. If $\lim_{x \to a} f(x)$ exists but $f(a)$ is undefined or different, the discontinuity is removable, a hole you could patch with one value. If both one sided limits exist and disagree, it is a jump. If either one sided limit is $\infty$ or $-\infty$, it is infinite and the graph carries a vertical asymptote there. If the function keeps oscillating with no settling value, as $\sin(1/x)$ does at $0$, it is oscillating.

Those four names cover the functions you will meet, not every function there is. Near $0$ the function $\sin(1/x)/x$ spikes to heights $\pi/2, 3\pi/2, 5\pi/2, \ldots$ with alternating signs and passes through $0$ between every pair of spikes, so neither one sided limit is infinite, and it is not bounded either. It fits none of the four boxes.

The type tells you what is still available. A removable discontinuity can be redefined at that point, so the function becomes continuous there, provided nothing else is broken elsewhere. A jump or an infinite discontinuity cannot be repaired by changing a single value. Either way the Intermediate Value Theorem and the Extreme Value Theorem need continuity across the whole closed interval, so one bad point means you can no longer invoke either theorem. The conclusion may still happen to hold, but nothing guarantees it.