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A piecewise function is usually defined by more than one formula: a fomula for each interval. Example 1: f( x ) = - x if x <= 2 = x if x > 2 What the above says is that if x is smaller than or equal to 2, the formula for the function is f( x ) = -x and if x is greater than 2, the formula is f( x ) = x.
Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the core concepts of topology, which is treated in full generality below.iven a sequence. of functions such that the limit. exists for all x in I, the resulting function f (x) is referred to as the pointwise limit of the sequence of functions (f n) nâN. The pointwise limit function need not be continuous, even if all functions f n are continuous, as the animation at the right shows.