Removing discontinuities is about fixing breaks in a function's graph so that it becomes continuous.
The mechanism involves redefining the function's value at a point where a limit exists but the function itself doesn't match that limit.
Imagine a piecewise function with a gap at x = a.
The left-hand and right-hand limits as x approaches a might exist and equal each other, but if the function value at a is different, the graph has a removable discontinuity.
To remove it, redefine the function at a to equal the common limit.
The cognitive trap lies in thinking that all discontinuities can be removed this way.
They can't.
Only when the left and right limits are equal and finite can you redefine the function to make it continuous.
Misunderstanding this leads to incorrect assumptions about continuity.
Also, watch out for endpoints in piecewise functions; they often hide discontinuities you might miss if you're not careful.