Removing discontinuities involves redefining a function at a point where a limit exists but the function is not currently defined or mismatches the limit.
This is about making the function's value at that point equal to the limit as x approaches the point.
The mechanism hinges on recognizing when a limit exists at a point of discontinuity: if it does, you can 'patch' the function by setting its value to this limit.
Piecewise functions often trip you up here; they require continuity at their domain boundaries, meaning the expressions on either side of the boundary and the function value at the boundary must all match.
The cognitive trap is assuming that if a function is piecewise, it's automatically discontinuous at the boundaries.
Not true.
You must check the limits from both sides and the actual function value.
Missing this means you might incorrectly label a function as discontinuous or fail to properly redefine it.