If a function is continuous, then it has no ‘breaks’ in the curve. Continuity helps in finding out if a function is predictable and ‘correct.’

Definition

Continuity

Let be a real-valued function (i.e. ). And let be a limit point of . Then:

Note, if is an isolated point, is always continuous at , with no condition.

Continuity (Alternative Definition)

Let and let be a real function . Let be a limit point of . Then:

Iff:

Proofs & Lemmas

Sums and products of continuous functions are continuous

Let be two real-valued functions. Let . Then:

Proof:

By doing algebra of limits (see tutorial!!)

Examples

  • : All constant functions are continuous
  • : The identity function is continuous