MT2002 Analysis
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Limits of functions

Our definition of continuity allows us to talk about the limit of a function.

Definition

If the sequence (f(xn) converges to the same limit for any sequence with (xn) rarrow p with xnnoteq p we call this limit limp f(x).

We can then rephrase the definition of continuity given above as:

A function f is continuous at p if limp f(x) exists and is equal to f(p).

Variations of the above definition are:

liminf f(p). [Take (xn) to be any unbounded monotonic increasing sequence.]

limminf f(p). [Take (xn) to be any unbounded monotonic decreasing sequence.]

limpp f(p). [Take (xn) to be any sequence converging to p with xn> p.]

limpm f(p). [Take (xn) to be any sequence converging to p with xn< p.]


Remark

  1. The condition xnnoteq p in the above definition is to allow for the possibility that f(x) is not defined at the point x = p. This is, for example, the case in the definition of the derivative of a function.
  2. In terms of quantifiers, we may define a function to be continuous if:
    (forallp belongs R)(forallepsilon > 0)(forallx belongs R)(thereexistsdelta > 0)(|x - p| < delta implies |f(x) - f(p)| < epsilon)
    Note that the value of delta that we need to find is allowed to depend on x as well as on epsilon.
    It is common for beginners to mis-state the definition as:
    (forallp belongs R)(forallepsilon > 0)(thereexistsdelta > 0)(forallx belongs R)(|x - p| < delta implies |f(x) - f(p)| < epsilon)
    In this case the same value of delta would have to work for all epsilon.

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    (The epsilon-delta definition)
    JOC September 2002