How to Handle a Hole and Other Discontinuities in a Function: Tips and Tricks

Hole at x=0 and 1 other point of discontinuity.

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A hole at x=0 is typically a removable discontinuity that occurs when a function is undefined at a certain point but can be made continuous by filling the hole with the value of the limit. To fill the hole, we can factor out the common factor from the numerator and denominator of the function expression that caused the hole and simplify.

For example, if we have the function f(x) = (x^2 – 4)/(x-2), there is a hole at x=2 because the denominator becomes zero. To fill this hole, we can factor out the common factor (x-2) from the numerator to get f(x) = (x-2)(x+2)/(x-2) = x+2. Therefore, the hole at x=2 can be filled with the value of f(2) = 4, and the function becomes f(x) = x+2 for all x except x=2.

If there is one other point of discontinuity, we need to determine what type of discontinuity it is and how to remedy it. There are three common types of discontinuities: jump, infinite, and oscillating.

A jump discontinuity occurs when the left-hand and right-hand limits of the function at the point of discontinuity are finite, but unequal. For example, the function f(x) = 1/x has a jump discontinuity at x=0 because the left-hand limit (approaching from negative x) is negative infinity and the right-hand limit (approaching from positive x) is positive infinity. Thus, we can fill the discontinuity with the limit value using the piecewise function f(x)={-1/x , for x<0 and 1/x, for x>0}.

An infinite discontinuity occurs when the left-hand or right-hand limit of the function at the point of discontinuity is positive or negative infinity. For example, the function g(x) = 1/x^2 has an infinite discontinuity at x=0 because the left-hand and right-hand limits approach infinity with opposite signs. In this case, we can try to simplify the expression or factor out any common factors to see if we can make the function continuous. If not, we can fill the discontinuity using a piecewise function, similar to the jump discontinuity method mentioned above.

An oscillating discontinuity occurs when the left-hand and right-hand limits of the function at the point of discontinuity exist, but the function oscillates between two or more values, approaching neither of them. A common example is the Dirichlet function, defined by f(x)={1, for x irrational and 0, for x rational}. It has oscillating discontinuities at every point.

In summary, to handle a hole at x=0 and one other point of discontinuity, we need to determine the type of discontinuity at the other point and fill each discontinuity using appropriate methods.

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