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Concept Version 9
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Continuity

A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output.

Learning Objective

  • Distinguish between continuous and discontinuous functions


Key Points

    • If a function is not continuous, it is said to be a "discontinuous function."
    • The function fff is continuous at some point ccc of its domain if the limit of f(x)f(x)f(x) as xxx approaches ccc through the domain of fff exists and is equal to f(c)f(c)f(c).
    • The function fff is said to be continuous if it is continuous at every point of its domain.

Terms

  • bicontinuous

    homomorphic or of structure-preserving mapping

  • topology

    a branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms


Full Text

A continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function." A continuous function with a continuous inverse function is called "bicontinuous." Continuity of functions is one of the core concepts of topology.

Example: Consider the function h(t)h(t)h(t), which describes the height of a growing flower at time ttt. This function is continuous. In fact, a dictum of classical physics states that in nature everything is continuous. By contrast, if M(t)M(t)M(t) denotes the amount of money in a bank account at time ttt, then the function jumps whenever money is deposited or withdrawn, so the function M(t)M(t)M(t) is discontinuous.

The function fff is continuous at some point ccc of its domain if the limit of f(x)f(x)f(x) as xxx approaches ccc through the domain of fff exists and is equal to f(c)f(c)f(c). In mathematical notation, this is written as limx→cf(x)=f(c)\lim_{x \to c}{f(x)} = f(c)lim​x→c​​f(x)=f(c).

In detail this means three conditions: 

  1. fff has to be defined at ccc,
  2. the limit on the left-hand side of that equation has to exist, and
  3. the value of this limit must equal f(c)f(c)f(c).

The function fff is said to be continuous if it is continuous at every point of its domain. If the point ccc in the domain of fff is not a limit point of the domain, then this condition is vacuously true, since xxx cannot approach ccc through values not equal to ccc.

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