# Continuity of Functions Exercises

#### 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...

#### Continuity on an Interval via Pictures

Remember, f is continuous on an interval if we can finger paint over f on that interval without lifting our drawing digit. Sample ProblemLook at the function f drawn below: The function f is...

#### 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'...

#### The Informal Version

Have a graphing calculator ready. Sample ProblemGraph the function f(x) = 2x. This is a polynomial, which is continuous at every real number. In particular, it's continuous at x = 4, wi...

#### The Formal Version

When we graph continuous functions, three things happen:We are given a continuous function f and a value c. We decide how far we wanted to let f(x) move away from f(c).We restrict the values of x...

#### 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)