| | We begin with a review of functions. Functions, in the most general sense, are rules which associate one thing with another. For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures. In elementary calculus, the domain and the range tend to be subsets of the real numbers. In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets. | | We begin with a review of functions. Functions, in the most general sense, are rules which associate one thing with another. For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures. In elementary calculus, the domain and the range tend to be subsets of the real numbers. In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets. |
| − | Functions which take a real numbers, and output a real number, are said to be '''continuous''' if, basically, they are connected. At right are illustrations of continuous and discontinuous functions. A strict definition of continuity is usually saved for a course in [[Real Analysis]]. | + | Functions which take a real numbers, and output a real number, are said to be '''continuous''' if, basically, they are connected. At right are illustrations of continuous and discontinuous functions. A strict definition of continuity is usually saved for a course in [[|real analysis|Real Analysis]]. |
| | The '''limit''' of a function is defined so that <math>\lim_{x\rightarrow x_0} f(x) = L \ </math> if, for any <math>\epsilon>0 \ </math>, there is an <math>\delta>0 \ </math> such that every number within <math>\delta \ </math> of <math>x_0 \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>. | | The '''limit''' of a function is defined so that <math>\lim_{x\rightarrow x_0} f(x) = L \ </math> if, for any <math>\epsilon>0 \ </math>, there is an <math>\delta>0 \ </math> such that every number within <math>\delta \ </math> of <math>x_0 \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>. |