Expert: Sherry Wallin - 5/29/2009

Question
Let F(x) = intergral (o,x) f(t) dt where f(x)= absolute value of (1-x) if absolute value of x is < or equal to 2 OR x^2 -1 if absolute value of x is > 2.
Find F. is it differentiable and continuous at x=2 and -2?

Answer
Hi Sarah~
    F(x) is not differentiable at x = +-2 thus cannot be continuous at either point. Check the limits as x -> 2 from both the left and the right and see that from the left of 2 the functional value of F(x) is 0 and from the right the functional value is 2/3 and since they are not equal F(x) at 2 is not differentiable. Likewise at x = -2, F(x) is not differentiable since the limit as x approaches -2 from the left is 4 and the limit as x approaches F(x) from the right the limit is -2/3 and 4 not equal -2/3.

Math Prof