# Continuity of Functions Examples

#### Continuity at a Point via Pictures

The Pencil Rule of ContinuityA continuous function is one that we can draw without lifting our pencil, pen, or Crayola crayon.Here are some examples of continuous functions:If a function is conti...

#### Continuity at a Point via Formulas

It's good to have a feel for what continuity at a point looks like in pictures. However, sometimes we're asked about the continuity of a function for which we're given a formula, instead of a pic...

#### Functions and Combinations of Functions

Many functions are continuous at every real number, x. These functions include (but are not limited to):all polynomials (including lines) ex sin(x) and cos(x)It's help...

#### Continuity on an Interval via Formulas

When we are given problems asking whether a function f is continuous on a given interval, a good strategy is to assume it isn't. Try to find values of x where f might be discontinuous. If we're a...

#### Continuity on Closed and Half-Closed Intervals

When looking at continuity on an open interval, we only care about the function values within that interval. If we're looking at the continuity of a function on the open interval (a, b), we don'...

#### Determining Continuity

When we say a function f is continuous, we usually mean it's continuous at every real number. In other words, it's continuous on the interval (-∞, ∞).Some examples of continuous functions tha...

#### Boundedness

The first theorem we'll attack is the boundedness theorem.Boundedness Theorem: A continuous function on a closed interval [a, b] must be bounded on that interval.What does mean to be bounded agai...

#### Extreme Value Theorem

We know. The title of this reading sounds pretty gnarly. The extreme value theorem, though, is just a slight extension of the boundedness theorem. There's really nothing all that extreme about it....

#### Intermediate Value Theorem

Intermediate Value Theorem (IVT): Let f be continuous on a closed interval [a, b]. Pick a y-value M, somewhere between f(a) and f(b)