Home > Calculus > Classifying Topics of Discontinuity (removable vs. non-removable)

What are the removable and non-removable discontinuities, if any, of #f(x)=x^2/absx#?

Answer 1

Charles Arocha

#f# has a removable discontinuity at #0#.

Recall that #absx = {(x, "if",x >= 0),(-x, "if",x < 0) :}#

So #f(x) = {(x^2/x, "if",x > 0),(x^2/-x, "if",x < 0) :}#

Notice that #f# is not defined for #x=0#, so #f# is not continuous at #0#.

We can simplify the definition of #f#, to get

#f(x) = {(x, "if",x > 0),(-x, "if",x < 0) :}#

This is simply the absolute value function except that it is left undefined at #0#.

We know (I think) that the absolute value function is continuous, so #f# is continuous everywhere except #0#.

Although #f(0)# is not defined, we can see that #lim_(xrarr0)f(x)# exists. (It equals #0#), so the discontinuity at #0# is removable.

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Answer 2

Scarlett Allison

The function f(x) = x^2/|x| has a removable discontinuity at x = 0 and a non-removable discontinuity at x = 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.