Question

If the functions \(f\) and \(g\) are continuous for all real \(x\), is \(f+g\) always continuous for all real \(x ?\) Is \(f / g\) always continuous for all real \(x ?\) If either is not continuous, give an example to verify your conclusion.

Short answer

The addition of two continuous functions \(f+g\) is always continuous, however the division \(f/g\) is not always continuous. For example, \(f(x) = x\) and \(g(x) = x^2\) are continuous, but \(f/g = 1/x\) is discontinuous at \(x = 0\).

Step-by-step solution

Analysis of addition

Firstly consider the addition of two functions. According to the property of limits, the sum of two continuous functions is also continuous. Therefore, for all real \(x\), \(f+g\) is always continuous.

Analysis of division

When it comes to division, we know that division by zero is undefined in mathematics. Therefore, if function \(g\) becomes zero at any point in its domain, the function \(f/g\) becomes discontinuous at that point since it will be undefined at that point.

Example

Let \(f(x) = x\) and \(g(x) = x^2\). Both of these functions are continuous for all real \(x\). However, the function \(f/g\) or \(x/x^2\) which simplifies to \(1/x\) is discontinuous at \(x = 0\). This supports our analysis that \(f/g\) is not always continuous.