Question

True or False It is possible to extend the definition of a function \(f\) at a jump discontinuity \(x=a\) so that \(f\) is continuous at \(x=a .\) Justify your answer.

Short answer

False. It is not possible to extend the definition of a function at a jump discontinuity so that it is continuous at that discontinuity.

Step-by-step solution

Understanding Definitions

First, we need to understand the key terms used in the exercise. A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Continuity at a point means that for any given tolerance, there exists a maximum allowable deviation of the independent variable such that the difference between the function value and the limiting value is within this tolerance. A jump discontinuity occurs when the function 'jumps' from one value to another value at a certain point in its domain.

Determining the Possibility of Redefinition

We can determine whether it is possible to redefine the function at the point of jump discontinuity. To make a function continuous at a point where it has a jump discontinuity, we must change the function value at that point to equate the left-hand limit and the right-hand limit. If the left-hand limit is not equal to the right-hand limit, they can't be made to coincide by changing just one value of the function, so in this case, it would not be possible to redefine the function to make it continuous at the point of discontinuity.

Justification

The justification is based on the definition of continuity. While a function can be redefined at points of removable discontinuity to make it continuous (by defining the function value to be the limit at that point), it cannot be redefined at points of jump discontinuity to make it continuous. This is because at a jump discontinuity, the left-hand and right-hand limits are different and we cannot adjust a single function value to make both the limits coincide.