A discontinuity is a point in a function where the function is either undefined, or is disjoint from its limit.

Jump discontinuity

A jump discontinuity occurs when right-hand and left-hand limits exist, but are unequal. That is:

$ \lim_{x\to a^+}f(x) \ne \lim_{x\to a^-}f(x) $

Removable discontinuity

A removable discontinuity occurs when left-hand and right-hand limits exist and are equal, but are undefined for the specified value.

Example: $ \lim_{x\to 0}\frac{sin x}{x} = 1 $

Infinite discontinuity

An infinite discontinuity occurs at points wehre the left-hand and right-hand limits are infinite.

Example: $ \lim_{x\to 0^+}\frac{1}{x} = \infty $ and $ \lim_{x\to 0^-}\frac{1}{x} = -\infty $

Other discontinuities

Other discontinuities exist, such as a value oscillating as it approaches a given value.

$ \lim_{x\to 0}\sin\frac{1}{x} $
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