Continuïteit van een functie

Consider a path along which a car moves. If the path is described by the curve of a function, the car’s position at any moment can be understood using the concept of a limit as the x-values approach a point on the curve.

As the car travels along the curve, smooth movement—meaning no breaks or jumps in path—suggests the function is continuous.

If two points approach the same point on the curve from opposite directions and meet smoothly, this indicates that the function is continuous at that point.

A curve can have a discontinuity at a point where the function is undefined, such as when division by zero occurs. As x gets closer to 1, the outputs approach 2. Since this limit exists, the discontinuity is removable, and the function can be redefined at that point to make it continuous.

If the left-hand and right-hand limits at a point don’t match, the function has a discontinuity at that point.

A curve that shoots upward or downward infinitely without touching a vertical line shows a vertical asymptote.

Sudden jumps in a piecewise function mark points where continuity breaks. These are called jump discontinuities.