Removing Discontinuities

A hole in a graph happens when a function is undefined or misdefined at one x-value, but the limit as x approaches that value still exists and is finite. The curve has a clear destination at that point, it just never lands there. That is completely different from a jump, where the function approaches two different heights from the left and right, or an asymptote, where it shoots off to infinity. A hole has one missing piece and one obvious shape to fill it with; a jump has no single value that works, because left and right disagree.

To remove the discontinuity, you redefine the function at that one x-value to equal the limit. If lim(x→c) f(x) = L but f(c) is undefined or different from L, define a new function g that matches f everywhere except at x = c, where g(c) = L. Now g is continuous at c: the limit, the function value, and the point on the graph all agree. Algebraically, holes usually show up as a factor that cancels. If f(x) = (x² - 9)/(x - 3), factoring gives (x-3)(x+3)/(x-3), which cancels to x + 3 for x not equal to 3. The limit as x approaches 3 is 6, so defining f(3) = 6 removes the hole.

This sits right before continuity rules and before derivatives, because derivatives require continuity, and limits of rational functions keep producing these cancelable holes. Recognizing a removable discontinuity versus a true break is the skill that lets you decide whether a function can be patched into something differentiable or whether it is broken for good.