Removing Discontinuities

A hole happens when a function is undefined at one point but the pieces around it still line up: the left-hand limit and right-hand limit agree. That shared limiting value is where the graph is clearly trying to go, so you can patch the hole by declaring the function to equal that value at that point. A jump discontinuity has no such agreement: the two sides head to different heights, so there's no single value that could fill the gap. That's the whole reason some discontinuities are removable and others aren't — removability depends on the limit existing, not on the function being defined.

Mechanically, you find the limit as x approaches the trouble point, usually by factoring and cancelling whatever is causing the 0/0 form, then plugging in. That simplified expression tells you the height of the hole. You then write a new piecewise definition: the original rule everywhere else, and the limit's value at the single point in question. The original formula itself never changes and its domain never changes — you're defining a new function that agrees with the old one almost everywhere and fills in the missing spot.

This sits right next to continuity itself: a function is continuous at a point only if it's defined there, the limit exists there, and the two match. Removable discontinuities are the case where two out of three already hold, so one new definition finishes the job. Jump and infinite discontinuities fail at the limit stage, which is why no single patch can fix them.