karl weierstraß gave another definition of continuity: imagine a function "f", defined on the real numbers. at the point formula_1 the function will have the value formula_2. if the function is continuous at formula_1, then for every value of formula_4 no matter how small it is, there is a value of formula_5, so that formula_6, means that formula_7. we can put this another way, given a point close to formula_1 (called x), the "absolute value" of the difference between the two values of the function can be made increasingly small, if the point x is "close enough" to formula_1.