IN MATH: 1.n. a break or missing point in the graph of a function. EX. The curve f(x) = sin(x)/x has a discontinuity when x is 0. See below.
IN ENGLISH: 1. as defined above.

Continuous Or Not Continuous 
Two piecewise defined functions are illustrated here. On the top graph, when x equals 0, the right half of the curve does not watch up with the left part of the curve. There is a discontinuity. On the bottom graph, even though the entire function is defined with 3 seperate functions, the pieces all match and create a continuous though "bumpy" curve. 
In Precalculus In precalc, one learns about polynomial function and rational function. Polynomial functions are always continuous. Rational functions often have discontinuities. Rational functions, a polynomial divided by a polynomial, a fraction with a polynomial in the numerator and in the denominator, have discontinuities. Since one cannot divide by zero, a zero in the denominator is "not legal" and results in a vertical asymptote. If the function has the same factor in the numerator and denominator a "hole" or missing point discontinuity exists. 
In Calculus I
In calc I, the first topic is
limits. In studying limits one considers the limit, as x goes to zero, of sin(x)/x, written as 
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