Type I discontinuities can be further distinguished as being one of the following subtypes: A jump discontinuity occurs when , regardless of whether is defined, and regardless of its value if it is defined. A removable discontinuity occurs when , also regardless of whether is defined, and regardless of its value if it is defined (but which does not match that of the two limits). A type II discontinuity occurs when either or does not exist (possibly both). This has two subtypes, which are usually not considered separately: An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote. An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits or does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include .
Type I discontinuities can be further distinguished as being one of the following subtypes: A jump discontinuity occurs when , regardless of whether is defined, and regardless of its value if it is defined. A removable discontinuity occurs when , also regardless of whether is defined, and regardless of its value if it is defined (but which does not match that of the two limits). A type II discontinuity occurs when either or does not exist (possibly both). This has two subtypes, which are usually not considered separately: An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote. An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits or does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include .