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`.