A function or curve is continuous if it is an unbroken curve. A more rigorous definition of continuity is:
A function is continuous at x if f(x) is defined and .
A function is continuous on a subdomain if it is continuous at all points in the subdomain. A function is continuous on the entire domain if it is continuous at all points in the domain. A function is discontinuous if it is not continuous. Click on the point on the slider in manipulative 1 and drag it to see various continuous and discontinuous functions.
There is a distinction between continuous on the left and continuous on the right. Take the ceiling function at x=1 as an example. At x=1, the ceiling function is continuous on the left, as the limit of f(a) as a approaches 1 from the left is 1. However, the limit of f(a) as a approaches 1 from the right is 2. To be continuous at x=1, f(x) must be continuous from the left and continuous from the right.
Continuity is important in analyzing functions, particularly in calculus. In calculus, one finds the derivative of a function. The derivative of a function exists only on subdomains that are continuous.
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