When it is difficult to find the limit of a function directly, it is sometimes possible to obtain the limit indirectly by "pinching" the function between simpler functions whose limits are known. For example, suppose if the lim xàa f (x) cannot be directly calculated, but we are able to find two functions, g and h, that have the same limit L as xàa and such that f is "pinched" between g and h by means of the inequalities g (x) W f (x) W h (x). It is clear that f (x) must also approach L as xàa because the graph of f lies between the graphs of g and h. Thus, the lim xàa g (x) = lim xà h (x) = L, since g and h have the same limit as x approaches a. The pinching theorem has applications to continually used limits of trigonometric functions; a) lim xà0 sin x = 0; lim xà0 cos x = 1; lim xà0 sin x / x = 1; lim xà0 1-cos x / x = 0.

When studying the concept of limits described above, a formal understanding of the theory of differentiation can be attained. Understanding differentiation graphically: Let the variable y be a function of the independent variable x, expressed by y = f (x). If x0 is a value of x in its domain of definition, then y0 = f (x0) is the corresponding value of y. Let h be a real number (Dx, "delta x,"), and let y0 + k = f (x0 + h);

A function which is not continuous at x = c is said to be discontinuous at that point or point of discontinuity. If we consider a function f (x) = {x+1, -2< x < 0; 2, x = 0; -x, 0 < x < 2; 0, x = 2; x - 4, 2 < x W 4, then we observe that f is not continuous at x = -2, 0 or 2. A discontinuity of the type found at x=0 is called a jump discontinuity. Although both left- and right-hand limits exist, they are different. The greatest-integer function, g (x)=[x] has a jump discontinuity at every integer. In the example, the discontinuity at x=2 is called removable.

If a function is continuous over an interval, its graph can be drawn lifting a pencil from the paper. The graph has no holes, breaks, or jumps on the interval. In addition, if the function is continuous at point c, then the closer x is to c, the closer f (x) gets to f(c). In order for any function to be continuous at c it must follow all three of the following constraints; a) f(c) must exist; that is, must be defined (c must be in the domain of f); b) lim xà c f (x) must exist; c) lim xàc f (x) must equal f(c). Also, a function is continuous under the interval [a, b] if it is continuous at each x such that a W x D b. When considering polynomials, they are continuous at every real number, and rational functions, P (x)/Q (x) are continuous at each point in their domain, that is, except where Q (x)=0. The function f (x)=1/x, for example, is continuous except at x = 0 , where f is not defined.

Some topics in this essay:
Axiom Induction, , Dx2y Dx2f, AC CB, Value Theorem, Dxy Dxf, = x0, = 0, derivative respect, function continuous, rate change, = x2, x =, independent variable, lim xà0, x0 +, = x0 +, x0 derivative respect, x = x0, addition function continuous, 1 lim xà0,