Hi Professor Puzzler. We learned in Calculus that if two functions are continuous, their sum is continuous. My teacher said that it doesn't work the other way - if the sum of two functions is continuous, that doesn't mean the two functions are continuous. She didn't give an example. Can you? Melinda, Arkansas

Hi Melinda,

The easiest way to give you a counter-example is using functions that are defined piecewise. Let f(x) = 1, when x > 0, and f(x) = 0 when x ≤ 0. Now let g(x) = 0 when x > 0, and g(x) = 1 when x ≤ 0,

The sum of these two functions is f(x) + g(x) = 1, for all values of x. This function is continuous, even though neither of the functions it was created from are continuous.

Here's another example: Let f(x) = [x] and g(x) = -[x]. Both of these are non-continuous functions (they are step functions), but when you add them, you get f(x) + g(x) = 0, which is also continuous.

As a matter of fact, if f(x) is any non-continuous function (but defined everywhere), doesn't it make sense that if g(x) = -f(x), the sum of the two functions would be continuous?