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