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Continuity of Functions
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 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...
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 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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