# 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 continu...

#### 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 are asked about the continuity of a function for which we're given a formula, instead of a pict...

#### 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 helpful...

#### 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 ask...

#### 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't i...

#### 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 that...

#### Boundedness

Boundedness Theorem: A continuous function on a closed interval [a,b] must be bounded on that interval.There are two numbers - a lower bound M and an upper bound N - such that every value of f on t...

#### Extreme Value Theorem

Maximum and Minimum ValuesThe maximum value of a function on an interval is the largest value the function takes on within that interval. Similarly, the minimum value of a function on an inter...

#### Intermediate Value Theorem

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

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