A continuous function is a function where small changes in the input result in small changes in the output. There are no breaks, jumps, or holes in its graph.
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A polynomial function is always continuous over all real numbers.
If a function is not continuous at a point, it is called discontinuous at that point.
The Intermediate Value Theorem applies to continuous functions, ensuring that they take on every value between any two values.
The sum, difference, and product of continuous functions are also continuous.
A rational function is continuous wherever its denominator is non-zero.
Review Questions
What is the definition of a continuous function?
How does the Intermediate Value Theorem relate to continuous functions?
Is the product of two continuous functions always continuous? Explain.
Related terms
Polynomial Function: A function that is represented by a polynomial equation and is always continuous over all real numbers.
Rational Function: A function defined by the ratio of two polynomials and may have discontinuities where its denominator equals zero.
Intermediate Value Theorem: A theorem stating that if a function is continuous on a closed interval $[a, b]$, then it takes on every value between $f(a)$ and $f(b)$.