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How to Determine Whether a Function Is Continuous

A graph for a function that's smooth without any holes, jumps, or asymptotes is called continuous. Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous at some value c in its domain:

  • f(c) must be defined. The function must exist at an x value (c), which means you can't have a hole in the function (such as a 0 in the denominator).

  • The limit of the function as x approaches the value c must exist. The left and right limits must be the same; in other words, the function can't jump or have an asymptote. The mathematical way to say this is that

    image0.png

    must exist.

  • The function's value at c and the limit as x approaches c must be the same.

    image1.png

For example, you can show that the function

image2.png

is continuous at x = 4 because of the following facts:

  • f(4) exists. You can substitute 4 into this function to get an answer: 8.

    image3.png

    If you look at the function algebraically, it factors to this:

    image4.png

    Nothing cancels, but you can still plug in 4 to get

    image5.png

    which is 8.

    image6.png

    Both sides of the equation are 8, so 'f(x) is continuous at x = 4.

If any of the above situations aren't true, the function is discontinuous at that value for x.

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