SOLUTION: Give an example of a function which is continuous at all points except -1,2 and 10.

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Question 1147481: Give an example of a function which is continuous at all points except -1,2 and 10.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


A rational function with factors of (x-a) in both numerator and denominator will have a "hole" in the graph at x=a -- that is, the graph will be continuous except at x=a.

The simplest (and not very interesting) function that is continuous everywhere except at x=-1, x=2, and x=10 is the rational function with factors of (x+1), (x-2), and (x-10) in both numerator and denominator:



Clearly that function has the value 1 everywhere that it is defined, which is everywhere except at x=-1, x=2, and x=10.

To get a more interesting function that satisfies the requirements, add any additional polynomial factors to the numerator. For example, if you add the factor (x^2-5) to the numerator, then the graph will look like the graph of x^2-5 but will have holes at x=-1, x=2, and x=10.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
For example, the function defined as


    h(x) = 0 everywhere, except  the points  x= -1, x= 2 and x= 10,

                         and equal to  1  at  x= -1, x= 2 and x= 10.


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