discontinous

Discontinuous, or discontinous in some spellings, describes something that is not continuous: a function, signal, or process that has breaks or jumps in its graph or behavior. In mathematics, discontinuity refers to a point at which a function fails to be continuous. A function f is discontinuous at a if either the limit as x approaches a does not exist or the value f(a) does not match that limit. Discontinuities are classified as removable, jump, and infinite (or essential). A removable discontinuity occurs when a function can be redefined at a to make it continuous, as in f(x) = (x^2 - 1)/(x - 1) for x ≠ 1, with f(1) defined appropriately. A jump discontinuity occurs when the left-hand and right-hand limits exist but are unequal, as in the Heaviside step function H(x). An infinite discontinuity occurs when the function grows unbounded near a, as in f(x) = 1/x near x = 0.

In analysis and topology, continuity is defined via limits or preimages of open sets; a discontinuity can thus be understood as a failure of these conditions at specific points. Discontinuities also appear in signals and time series as abrupt changes, caused by phenomena such as shocks or measurement errors. The term is largely synonymous with non-continuous, although in some contexts "discontinuous" is preferred over "discontinous." See also continuity, limits, and types of discontinuities.