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 xnp we call this limit f (p).

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 xnp 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:
(∀pR)( ε > 0)(∀xR)(∃δ > 0)(|x - p| < δ ⇒ |f (x) - f (p) < ε)
Note that the value of δ that we need to find is allowed to depend on x as well as on ε.
It is common for beginners to mis-state the definition as:
(∀pR)(∀ε > 0)(∃δ > 0)(∀xR)(|x - p| < δ ⇒ |f (x) - f (p) < ε)
In this case the same value of δ would have to work for all ε.

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JOC September 2001