Limits: Discontinuity, a function with ‘breaks’

In Calculus, a branch of mathematics we learn about limit of a function, we say that the limit of f(x) is L as x approaches ‘a’ and is written as lim(x->a) f(x)=L, provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.

Discontinuity : A function is said to be continuous at x=a if lim(x->a)f(x) = f(a). In simple words a function is said to be continuous if the graph of the function has no breaks in it, that is, it is a continuous curve. Many functions, however, will have isolated points where they are not connected. Such type of points is called Discontinuity points of that function. Definition of Discontinuity can be given as, a function is discontinuous at ‘a’ if it is defined at ‘a’ but is not continuous at ‘a’. Discontinuity points can be classified into three types.

The function f(x) has a discontinuity of first kind at x=a if there exists left hand limit, lim(x->a-0)f(x) and right hand limit (x->a+0 f(x) and also these one-sided limits are finite.
Point discontinuities are also called the removable discontinuities. Sometimes we come across functions that are defined differently for a certain point. Let us consider the function f(x)=1 for x=3 and f(x) =x^2,  for all positive real numbers. We define the value of the function to be one at the point x=3, though the rest of the function is given by f(x) = x^2. When we graph this function, we can see that the function is continuous except for this tiny hole in the curve at x=1. It is discontinuous at a single point x=1, and this discontinuity is called the point discontinuity or the removable discontinuity. In general, point discontinuities occur when a function is defined specifically for an isolated value of x.

What is Jump Discontinuity
One might wonder what is a Jump Discontinuity. Let us now learn about a Jump Discontinuity, in this type of discontinuity the right hand limit is not equal to the left hand limit of a function f(x). Jump discontinuities are also called simple discontinuities. Let us consider a function, f(x) = x^2 for x less than equal to 1 and f(x) =6-x^2 for x greater than 1. The two pieces have different value at x=1, and the graph of these function f(x) seem to ‘jump’ from one branch to another and this jump makes the function discontinuous. We refer this discontinuity as a jump discontinuity

How to do Regression Testing

Regression Testing
Regression in statistics is the relation between selected values of x and observed values of y from which the most probable value of y can be predicted for any value of x. For example, a researcher in the medical field might want to use the age of a person (independent variable) to predict the most appropriate dose for a new drug (dependable variable) The purpose of running the regression is to find the formula that fits the relationship between the two variables. This regression formula helps to predict values for the dependent variable when only the independent variable is known. The end result of the analysis or testing would help a doctor to prescribe the proper dose based on person’s age.  The analysis done along with test procedures involving finding the standard error of the slope, degree of freedom, test statistic (t-score) and p-value which help in interpreting the results of the testing  whether the standard requirements for a simple linear regression are satisfied or not is known as Regressing testing.

Regression Testing Definition
We can define Regression Testing as a statistical tool that investigates the relationship between the dependent and independent variables in a linear regression.
Consider a Regression Testing Example: Using the sample data, determine the slope standard error, the regression line slope, the degree of freedom, the statistic test, and also the P-value accompanying the test statistic. The data given below is a output formed hypothetically for the regression equation y’= 76 +35x [y’= a+bx, b is the slope]

Predictor        Coefficient Standard error coefficient T P
constant                76 30 2.53       0.01
  x 35 20 1.75       0.04

Step1: calculate the slope’s standard error, and for this we use the formula,
SE = sqrt{sigma(y1- y’1)2/(n-2)} divided by {sqrt[sigma(xi- x(bar))2]
xi  is the observed values, x(bar) is considered as the mean of the independent variables, for number of observations we denote with n
Step2: The regression line slope is 35.
Step3: Degree of freedom is calculated using, DF = n-2, where n is the number of observations.
Step4: Test static is a t-score (t) given by t=slope of the regression/standard error of the slope.
Step5:  the probability of observing a sample statistic as extreme as the test static is considered as P value.
A t-Distribution Calculator is used to evaluate the probability considered with the test statistic
using the computed degree of freedom.
Step6: Finally the results are interpreted. Comparing the P-value with the significant level (which is
            mostly taken between 0 and 1) and the null hypothesis is rejected when the P-value is found to be less than the significant level.

Know more about the statistic help, Math Homework Help. This article gives basic information about Regression Testing. Next article will cover more statistic concept and its advantages,problems and many more. Please share your comments.