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# Continuity - Problem 3

FREE###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

To check if a function is continuous at a point, first check if the function exists at that point (if for that x, there exists an f(x)). Then, check if the limit as x approaches that point exists. Recall that this can be done by checking if the left- and right-hand limits are the same. Finally, for a function to be continuous, the value of the function at a point must be the same as the limit of the function at that point.

Let's do one last example. Why is this function; y equals h(x), I have graphed here not continuous at x equals 4? Recall the definition of continuity.

There are three conditions. The function's got to be defined at the number in question in this case 4. The limit as x approaches that number of f(x) has to exist. The limit as x approaches a of f(x) has to equal the value at a.

Here, the function is actually defined at 4. This point indicates that f(4) is 5. The limit exists as well. As you approach 4 from either the right, or the left, the limiting value is 1. The problem is that these two things are not the same. The value of the function is 5. The limit of the function is 1.

So the problem is, that the limit as x approaches 4 of h(x) is 1. That's not equal to h(4) which is 5. So this violates condition 3 of the definition of continuity.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

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