The Continuity of the Product of Functions: A Proof and Considerations

When dealing with the theory of functions in real analysis, one often encounters questions about the continuity of the product of two functions. This article aims to explore the conditions under which the product of two functions, f and q, is continuous at a point c. Specifically, given a function f defined on a non-empty proper subset of R that is continuous at a point c where fc 1, and a bounded function q defined on the same set, we will investigate whether the product fq is always continuous at c.

Introduction to the Problem

Consider the question: if f is a function defined on a non-empty proper subset of R, which is continuous at a point c, and qc 1, and q is a bounded function also defined on the same set, is it always true that the product fq is also continuous at c? The answer is no, and we will provide counterexamples to demonstrate this.

Counterexample I: Bounded Function Discontinuous at c

To illustrate why fq is not always continuous at c, let's construct a specific counterexample. Suppose we have:

Case 1: c 0

Define fx 1 - 1/x for x > 0, and fx -1 for x x and satisfies fc 1 at c 0.

Define qx signum(x), which is the sign function that returns 1 for positive x, -1 for negative x, and 0 at x 0. This function is bounded but discontinuous at c 0.

The product function fq q, which is not continuous at c 0 since qc 1 and qx switches from -1 to 1 at x 0. This example shows that the product of a continuous function and a bounded discontinuous function can still be discontinuous.

Counterexample II: A General Framework

Let's formulate a more general approach to demonstrate that the product fq is not always continuous at c.

If we require q to be bounded and discontinuous at c, we can construct such a function. Consider a discontinuous function such as q(x) (x - c) / |x - c|. However, this function is not bounded as it approaches infinity near c. Thus, we need a bounded variant of this discontinuous function.

One such bounded function could be: qx signum(x - c). This function is discontinuous at x c and bounded, satisfying the required conditions. For fx 1 for all x, the product fq q(x) is discontinuous at c.

Conclusion and Further Considerations

To summarize, the product of two functions, where one is continuous and bounded and the other is bounded and discontinuous at a specific point, may not be continuous. The key factor is the discontinuity of the bounded function.

To prove this in more detail, we can use the limit definition of continuity. If fc 1, then for any sequence x_n converging to c, the product fx_n qx_n is simply qx_n. Since q is not continuous at c, the limit of qx_n as x_n approaches c does not exist, leading to the discontinuity of fq at c.

Therefore, when analyzing the continuity of the product of two functions, it is crucial to consider the continuity properties of both functions, particularly the bounded and discontinuous functions involved. This knowledge is vital for advanced topics in real analysis and functional analysis.

Keywords: continuity, product of functions, bounded functions