continuous function

A function on the set of real numbers is considered to be continuous if its graph on the Cartesian plane is a smooth curve with no “holes” or “jumps.” In other words, any arbitrarily small change to the function’s input corresponds to an arbitrarily small change in its output.

A function is said to be continuous at a point, if there is no hole or jump at that point. If there is, the point is called a discontinuity. So, for a function to be continuous, it must be continuous at every point in its domain.

This definition is usually sufficient to judge a function’s continuity, but it is not mathematically rigorous. More formally, we can say that if f is function mapping I, a subset of the real numbers, to R, the set of real numbers:

f: I \mapsto R

Then is continuous at a point c, if the limit of f(x) as x approaches c exists and is equal to f(c). In mathematical notation:

\lim_{x\to{c}}f(x) = f(c)

There are 3 rules implicit in this definition:

  1. f is defined at c,
  2. the limit as x approaches c from the lefthand side of the domain must exist,
  3. and that limit must equal f(c).
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