Continuity limit graph help

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Originally Posted by emakarov

Do you mean you have (1,0) in the graph? The jump discontinuity is called this way not because there is a jump between the limit of the function (1 in this case) and the value of the function (0 in this case) at that point. The fact that there is a discontinuity already means that either the limit does not exist or it is not equal to the function value. Rather, it is called jump discontinuity because there are left and right limits and there is a jump between those. The value of the function does not matter. In this case, both left and right limits are 1, so this is a removable discontinuity (if f(1) were 1 instead of 0, the function would be continuous at x = 1).
c = 0 is correct, but the limit when x -> 1 exists.
Do you mean, "two possibilities"?

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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.

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Originally Posted by fkf

For the second question we need to know that

lim{x->a} f(x) = L <=> lim{x->a+} f(x) = lim{x->a-} f(x) = L

to consider the function having a limit L at x = a.

Thus we have no limit att x = 0 and neither at x = 1.

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Why is there no limit at x = 1?

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Originally Posted by emakarov

Why is there no limit at x = 1?

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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.

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Originally Posted by fkf

Plot a graph

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Like this?

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.