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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
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?