continuous function is a function without the "jumps", ie one for which the condition: small changes in the argument followed by small changes in the values of the respective functions.The graph of such a function is a smooth and continuous curve.
continuity at a point to a set limit can be determined using the concept of the limit, namely, the function should have a limit at this point, which is equal to its value at the limit point.
When these conditions at some point, saying that the function at this point is discontinuous, that is, its continuity is broken.In the language of limits break point can be described as the difference in the values of the bursting point with a limit function (if it exists).
break point can be removable, it is necessary that the limit function, but it does not match the value at a given point.In this case, at this point it is possible to "correct", i.e. to extend the definition of continuity.
completely different picture emerges if the limit of a function at a given point does not exist.There are two possible points of discontinuity:
- first kind - are finite and both of the one-sided limits, and the value of one or both of them do not coincide with the value of the function at a given point;
- second kind, where there is one-sided or both of the limits or values endless.
properties of continuous functions
- function resulting from arithmetic operations, as well as composition of continuous functions on their domain is also continuous.
- Given a continuous function which is positive at some point, you can always find a sufficiently small neighborhood in which it will retain its character.
- Similarly, if the values of the two points A and B are, respectively, a and b, wherein a is different from b, then for the intermediate points, it will take all the values in the interval (a; b).From here you can make an interesting conclusion: if you give a stretched rubber band to shrink so that it does not sag (remained straight), one of its points will remain fixed.A geometrically it means that there is a straight line passing through any intermediate point between A and B, which intersects the graph of the function.
note some of the continuous (in the domain of definition) of elementary functions:
- constant;
- rational;
- trigonometry.
between the two fundamental concepts in mathematics - is continuous and differentiable - are inextricably linked.It is enough to recall that for differentiable functions you need it to be a continuous function.
if the function is differentiable at some point, there is continuous.However, it is not necessary, so that its derivative is continuous.
features available on some set of continuous derivative, belongs to a separate class of smooth functions.In other words, it is - a continuously differentiable function.If the derivative has a limited number of break points (only the first kind), then a similar function called piecewise smooth.
Another important concept of mathematical analysis is uniformly continuous functions, that is, its ability to be at any point in its domain equally continuous.Thus, a property that is considered at a plurality of points rather than a single.
If you fix a point, you get nothing else, as the definition of continuity, that is, from the existence of uniform continuity it follows that this is a continuous function.Generally speaking, the converse is not true.However, according to Cantor's theorem, if a function is continuous on the compact, that is, on a closed interval, then it is uniformly continuous on it.
