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 pictu...
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 c...
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't...
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, with...
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 u...
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 again?...
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)