I've been wondering, how do you exactly prove that a function is continuous everywhere (or within the domain in which the function is defined)? Given some curve, my current approach would be to to try to think of discontinuities and then find a single counter-example.
But I can imagine that this method has its own limitations for strange functions.
Is there a set 'method' or 'technique' to prove that a function is continuous?
My second question is involving the sum of two discontinuous functions.
If I have a curve with point discontinuity (where at that point, it takes an arbitrary value not equal to the left and right limit), I can easily find another discontinuous curve that when added together, forms a continuous function.
However, lets say the discontinuity is because there is an actual 'hole' in the number line, so for example $y=\frac{x^2-1}{x-1}$, is it possible to find another discontinuous function such that upon addition, the sum is continuous?
My gut is telling me that the answer is 'No' because if the point didn't exist in the first place, then it cannot suddenly appear out of nowhere. But I would like some confirmation with this.
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