Understanding the Continuity of Functions and Their Sum

Understanding the relationship between the continuity of a sum of functions and the individual continuity of those functions is crucial in calculus and analysis. A common question in this context is whether the sum of two functions being continuous at a point a implies that both individual functions are also continuous at a. This article explores this query by providing a detailed analysis and several counterexamples that clarify the nuances of function continuity.

Definition of Continuity

A function h(x) is continuous at a point a if and only if the limit of h(x) as x approaches a equals h(a). Mathematically, this condition can be stated as:

[ lim_{{x to a}} h(x) h(a) ]

Continuity of the Sum of Functions

To check if the sum of two functions, f(x) g(x), is continuous at a point a, we examine the limit of the sum as x approaches a and compare it to the sum evaluated at a. If both f(x) and g(x) are continuous at a, then their sum is also continuous at a. However, the converse is not always true. Even if the sum is continuous at a, it does not necessarily mean that both f(x) and g(x) are continuous at a.

Counterexample

Consider the following functions:

f(x) 0 for all xg(x) 1 if x ≠ a and g(x) 0 if x a

Note that f(x) is continuous everywhere, but g(x) is not continuous at a. However, their sum, h(x) f(x) g(x), is continuous at a because:

[ lim_{{x to a}} (f(x) g(x)) lim_{{x to a}} 0 1 1 0 0 f(a) g(a) ]

More Counterexamples

There are multiple ways to provide counterexamples to demonstrate that the individual continuties of f(x) and g(x) are not required for f(x) g(x) to be continuous at a.

Counterexample 1

Let f(x) 1 if x is rational and 0 if x is irrational.

Let g(x) 0 if x is rational and 1 if x is irrational.

Then, f(x) g(x) 1 for all x. Hence, f(x) g(x) is everywhere continuous, whereas both f(x) and g(x) are nowhere continuous.

Counterexample 2

Given any function f(x) that is not continuous at a, set g(x) -f(x). The sum f(x) g(x) 0 is constant, which is obviously continuous everywhere, even though f(x) is discontinuous at a.

Counterexample 3

Define function f(x) as follows:

fx a if x a/b in Q where aab are integers with a ∨ b 1, otherwise fx 0

For this function, f(x) is discontinuous everywhere. If we define g(x) -f(x), then f(x) g(x) 0 is continuous everywhere, reinforcing the idea that the sum can be continuous while the individual functions are discontinuous.

Counterexample 4

Consider two functions:

fx 1 if x 0, fx -1 if x 0gx -1 if x 0, gx 1 if x 0

Note that both fx and gx are discontinuous but their pointwise sum fx gx 0 is continuous throughout the real line.

Conclusion

In conclusion, while the continuity of the sum f(x) g(x) at a point a provides some indication of the behavior of the individual functions, it does not guarantee their continuity. Understanding these nuances is essential for grasping the deeper concepts in analysis and calculus.