Which functions are continuous

Removable discontinuities are those where there is a hole in the graph as there is in this case. A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it.

Rational functions are continuous everywhere except where we have division by zero. So all that we need to is determine where the denominator is zero. With this fact we can now do limits like the following example. Below is a graph of a continuous function that illustrates the Intermediate Value Theorem. Also, as the figure shows the function may take on the value at more than one place. It only says that it exists. These are important ideas to remember about the Intermediate Value Theorem.

A nice use of the Intermediate Value Theorem is to prove the existence of roots of equations as the following example shows. For the sake of completeness here is a graph showing the root that we just proved existed.

Note that we used a computer program to actually find the root and that the Intermediate Value Theorem did not tell us what this value was. If it does, then we can use the Intermediate Value Theorem to prove that the function will take the given value. We now have a problem. 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 :. Later you will meet the concept of differentiation. We will learn that a function is differentiable only where it is continuous.

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