When is a graph discontinuous

Since the function doesn't approach a particular finite value, the limit does not exist. This is an infinite discontinuity. Notice that in all three cases, both of the one-sided limits are infinite.

These holes are called removable discontinuities. Removable discontinuities can be fixed by redefining the function, as shown in the following example. Next, using the techniques covered in previous lessons see Indeterminate LimitsFactorable we can easily determine.

Now we can redefine the original function in a piecewise form:. The first piece preserves the overall behavior of the function, while the second piece plugs the hole. Rather than define whether a function is continuous or not, it is more useful to determine where a function is continuous. To prove a function is not continuous, it is sufficient to show that one of the three conditions stated above is not met.

When a function is not continuous at a point, then we can say it is discontinuous at that point. There are several types of behaviors that lead to discontinuities. A removable discontinuity exists when the limit of the function exists, but one or both of the other two conditions is not met. The graphical feature that results is often colloquially called a hole. This type of function is frequently encountered when trying to find slopes of tangent lines.

There are some functions that are not defined for certain values of x. We can see that there are no "gaps" in the curve. Any value of x will give us a corresponding value of y. We could continue the graph in the negative and positive directions, and we would never need to take the pencil off the paper.

In simple English: The graph of a continuous function can be drawn without lifting the pencil from the paper. Many functions have discontinuities i.

We see that small changes in x near 0 and near 1 produce large changes in the value of the function. There are 3 asymptotes lines the curve gets closer to, but doesn't touch for this function.

Note: You will often get strange results when using Scientific Notebook or any other mathematics software if you try to graph functions which have discontinuities. It is showing us all the vertical values that it can from an extremely small negative number to a very large positive number - but we can't see any detail certainly none of the curves.

We need to restrict the y -values so we can see the true shape of the curve, like this I have changed the view of the vertical axis from to 10 :.

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