# List of Notes - Category: Continuity of the Function

## Definition of Continuous Function

Definition. A function f is continuous at a if lim_(x->a)=f(a).

Continuity implies three things:

1. f(a) is defined (i.e. a is in domain of f)
2. lim_(x->a)f(x) exists
3. lim_(x->a)f(x)=f(a)

Geometrically continuity means that you can draw function without removing pen from the paper.

## One-Sided Continuity. Classification of Discontinuities

Similarly to the one-sided limits, we can define one-sided continuity.

Definition. Function f(x) is continuous from the right at point a if lim_(x->a^+)=f(a). Function f(x) is continuous from the left at point a if lim_(x->a^-)f(x)=f(a).

## Theorems involving Continuous Functions

Intermediate Value Theorem. Suppose that f is continuous on closed interval [a,b] and let N is any number between f(a) and f(b) (or f(b) and f(a); depends what is bigger). Then there exists number c in (a,b) such that f(c)=N.