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) p with xn p we call this limit f(x).

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

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

Variations of the above definition are:

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

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

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

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

Remark

1. The condition xn 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:
(p R)( > 0)(x R)( > 0)(|x - p| < |f(x) - f(p)| < )
Note that the value of that we need to find is allowed to depend on x as well as on .
It is common for beginners to mis-state the definition as:
(p R)( > 0)( > 0)(x R)(|x - p| < |f(x) - f(p)| < )
In this case the same value of would have to work for all .

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JOC September 2002