Question

True or False? In Exercises 61-64, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(g\) and \(h\) are continuous functions of \(x\) and \(y\), and \(f(x, y)=\) \(g(x)+h(y),\) then \(f\) is continuous.

Short answer

The statement is true.

Step-by-step solution

Understand the definition of continuity

A function \(f(x)\) is said to be continuous at a certain point \(x=a\) if \(lim_{x\to a} f(x) = f(a)\). If this condition holds true for all the points in the domain of \(f(x)\), the function is said to be continuous everywhere. The key here to note is that both \(g(x)\) and \(h(y)\) are continuous functions.

Comprehend the behavior of continuity upon function addition

Following the algebra of limits, if \(g\) and \(h\) are continuous at a point \(x=a\), then their sum, i.e., \(f(x) = g(x) + h(x)\) should also be continuous at \(x=a\). This indicates that if \(g\) and \(h\) are continuous then their sum function \(f(x,y)=g(x)+h(y)\) should also be continuous.

Determine the statement's truthfulness

Given that \(g(x)\) and \(h(y)\) are continuous and utilizing the behaviors of continuity upon function addition, it can be concluded that the function \(f(x,y) = g(x) + h(y)\) is also continuous. Therefore, the given statement is true.