Question

True or false: If f: R → R is discontinuous at 0 we can always redefine it at zero to make it continuous, that is, there exists a g: R → R such that g(x) = f(x) for all nonzero r and g is continuous at 0.

Answer 1

It is true that a function f: R → R that is discontinuous at 0 can often be redefined at 0 to make it continuous by choosing g(0) to be the limit of f(x) as x approaches 0, assuming the limit exists and is finite.

Step-by-step explanation:

The statement is true. If a function f: R → R is discontinuous at 0, it is indeed possible to redefine it at zero to make it continuous by choosing an appropriate value for g(0).

The function g: R → R is defined such that g(x) = f(x) for all non-zero x, and to ensure continuity at x = 0, we can define g(0) to be the limit of f(x) as x approaches 0, provided the limit exists. If this limit does not exist, then defining g(0) to make g continuous at x = 0 will not be possible.

This concept relates to the idea that a function y(x) must be a continuous function and the first derivative of y(x) with respect to space, dy(x)/dx, must also be continuous for the function to be considered continuous.