You know it never occurred to me that the function would not have a discontinuity at .
But I have reviewed about ten different calculus texts, and they are about evenly split on what it means. What did not occur to me is discontinuity may not mean not continuous at
We say a function has property if and only if
So a function does not have property if and only if .
So once again, read your textbook. You may not have missed points on that question.
Plot a graph and then take the given limits from both sides and se that they're not the same.
lim{x->1+}f(x) =/= lim{x->1-}f(x)
We can say that we have two different onesided limits there. But usually when talking about "limit" then left+rightlimit needs to be the same.
Yep, and then One-sided limit - Wikipedia, the free encyclopedia
Look, are we speaking different dialects of English and that's why we don't understand each other? Are you talking about the limits of f' or of f itself? Aren't both one-sided limits equal to 1? On the Desmos graph you can even click the plot and move the point along the graph, and it will show you the coordinates.
In post #15 you wrote that we have a removable discontinuity at x = 1. But this means (Wikipedia) that both one-sided limits are equal. Indeed 1/x = 1 when x = 1: this is the left limit. On the right of x = 1, f(x) = 1, so the right limit is also 1.