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Continuity of Functions
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Calculus
Continuity of Functions
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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 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 HalfClosed 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 yvalue M with f(a)
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