Discontinuity points. Who was born first? The egg or the chicken?

Hi I started doing the "Calculus in a Nutshell" because even if I passed the exam at university I still have a lot of doubts about my understanding. So when I answered this question I had a new doubt. As you can see the Function since has x at denominator has to have a point of discontinuity for x=0. But since this function is equivalent to a straight line if I only see the straight line function I should have no discontinuity points. So my question is... If someone shows me a function like the one in the answer and ask me if there are discontinuity points and I answer that there are none I am wrong because I can lead back the function to the one with a denominator? Or it's just a matter of who was born first? The egg or the chicken?

I apologize for my bad English.

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4 months, 1 week ago

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Is it possible to give down the function you had difficulties with? (You haven’t given the function in the note)

Jason Gomez - 4 months, 1 week ago

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I posted a screenshot in the post but I'll write it down anyway the function is f(x)=(x+a)2a2xf(x)=\frac{(x+a)^2-a^2}{x} can be rewritten as f(x)=x+2af(x)=x+2a with x0x\neq0

- - - 4 months, 1 week ago

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Oh sorry the screenshot didn't load for me even after I had read your question twice.

Jason Gomez - 4 months, 1 week ago

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If someone were to show you the function f(x)=x+2af(x) = x + 2a, you would certainly be correct in saying that there are no discontinuity points. However, if someone were to show you the function f(x)=x+2a, x0f(x) = x + 2a, ~\boldsymbol{x \neq 0}, then you would not be able to give a correct answer. Whoever was showing you this function would be guilty of not giving you enough information. In other words, they would be asking you a question about all points on f(x)f(x), but not telling you what happens when x=0x = 0. So it would be impossible to give an answer. You could perhaps guess that there was a discontinuity point, since they told you that f(x)f(x) was only correct when x0x \neq 0, which is "suspicious". But the only way to indicate that there is a discontinuity at x=0x = 0 is to give the full function, f(x)=(a+x)2a2xf(x) = \frac{(a+x)^2 - a^2}{x} . Hope that helps!

David Stiff - 4 months, 1 week ago

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Yup it helped. Thank you very very much!

- - - 4 months, 1 week ago

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No problem!

David Stiff - 4 months, 1 week ago

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