How to investigate the continuity of function

How to investigate the continuity of function

Continuity - one of the main properties of functions. The decision on that, this function is continuous or not, allows to judge other properties of the studied function. Therefore it is so important to investigate functions on continuity. In this article the main receptions of a research of functions on continuity are considered.


1. So, we will begin with determination of continuity. It says the following: The f (x) function defined in some vicinity of a point of an is called continuous in this point, eslilim f (x) =f(a)x-> a

2. Let's understand, what does it mean. First, if functionnot definedthere is no in the let point, then sense speak about continuity. Function of a razryvn and point. For example, all the known f(x)=1/x does not exist in zero (it is impossible to divide into zero at all), here and a gap. Same will concern also more difficult functions in which it is impossible to substitute some values.

3. Secondly, there is other option. If we (or someone for us) composed function from pieces of other functions. For example, such: f(x) =x^2-4, x <-13x,-1 <=x<35, x> = 3B such case to us it is necessary to understand, it is continuous or a razryvna. How to make it?

4. It is option more difficult as it is required to establish continuity on all range of definition of function. In this case a range of definition of function is all numerical axis. That is from minus infinity to plus infinity. For a start we will use determination of continuity on an interval. Here it: The f (x) function is called continuous on a piece [a; b] if it is continuous in each point of an interval (a; b) and, besides, it is continuous on the right in a point of an and at the left in b point.

5. So, to define the continuity of our difficult function, it is necessary to answer for himself several questions: 1. Whether the taken functions on the set intervals are defined? In our case affirmative answer. Means, points of a gap can be only in function change points. That is in points-1 and 3.

6. 2. Now it is necessary to investigate the continuity of function in these points. We already know how it becomes. At first it is necessary to find values of function in these points: f (-1)=-3, f(3)=5 - function is defined in these points. Now it is necessary to find the right and left limits for these points.lim of f (-1)=-3 (the limit at the left exists) x->-1-lim f (-1)=-3 (the limit exists on the right)>-1+ As we see x-, the right and left limits for a point-1 coincide. Means, function is continuous in a point-1.

7. Let's do the same for a point 3.lim f(3)=9 (the limit exists) x-> 3-lim f(3)=5 (the limit exists) x-> 3+ And here limits do not coincide. It means that in a point the 3rd function of a razryvn. That's all research. We wish success!

Author: «MirrorInfo» Dream Team