Are There Functions Discontinuous Individually but Continuous When Combined?

[semidevil]

I"m trying to think of 2 functions that are discontinuous at point C, but when added togther, and multiplied togther, will be continuous at point c.

I tried 1/x, root(x), a polynomail w/ x-1 in the denomiator...cant think of anything...any hints?

I mean, when you multiply 2 functions to get a new function, can't you factor that function back to the original function, and it will be discontinuous again?

 

[Data]

Think quite a bit simpler. Hint: Try two functions that are "almost constant," ie. they are constant except at $C$ where they are both discontinuous.

Don't forget that defining functions piecewise is perfectly allowable.

 

[thepatientmental]

It is possible to find two functions that are individually discontinuous at point C, but when added and multiplied together, they become continuous at point C. One example is the function f(x) = 1/x and g(x) = x. Individually, both functions are discontinuous at x = 0, but when added and multiplied together, the resulting function h(x) = f(x) + g(x) = 1 + x is continuous at x = 0. Another example is the function f(x) = sqrt(x) and g(x) = -sqrt(x). Individually, both functions are discontinuous at x = 0, but when multiplied together, the resulting function h(x) = f(x) * g(x) = -x is continuous at x = 0.

As for your question about factoring the function back to the original and it becoming discontinuous again, this is not always the case. In the examples given above, even if we factor the resulting function back to the original functions, they will still remain continuous at point C. This is because when we multiply two functions, we are essentially finding the point-wise product of the functions, which can behave differently than the individual functions. This is why it is possible to find functions that are discontinuous individually, but become continuous when multiplied together.