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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)
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