§ 18 One of the most basic features of a function is...

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Math 132 Continuity Stewart § 1.8 One of the most basic features of a function is whether it is continuous . Roughly, this means that a small change in x always leads to a fairly small change in f ( x ), without instantaneous jumps. In physical terms, the position of a particle moving in space is continuous, but the position displayed in a video could have a gap, making the position function jump discontinuously. This can be made precise by saying that near x = a , the limit of f ( x ) is f ( a ): Definition: A function f ( x ) is continuous at x = a whenever lim x a f ( x ) = f ( a ). Graphically, a function is continuous whenever the graph y = f ( x ) proceeds through the point ( a, f ( a )) without jumps or holes. Types of discontinuity. If f ( x ) is defined near x = a , continuity can fail in several ways: i. Removable discontinuity: f ( a ) is undefined, but lim x a f ( x ) exists. ii. Removable discontinuity: