Antiderivative of discontinuous function I am having confusion regarding anti-derivative of a funct

Marissa Singh

Marissa Singh

Answered question

2022-05-10

Antiderivative of discontinuous function
I am having confusion regarding anti-derivative of a function.
f ( x ) = { x 2 2 + 4 x 0 x 2 2 + 2 x > 0
Consider the domain [-1,2]. Clearly the function is Riemann integrable as it is discontinuous at finite number of point. However is there a function g(x) such that g ( x ) = f ( x ) x [ 1 , 2 ]?

Answer & Explanation

rynosluv101swv2s

rynosluv101swv2s

Beginner2022-05-11Added 19 answers

Step 1
No. The best you can do is a function g that is continuous everywhere, and differentiable with g ( x ) = f ( x ) for all x except at x = 0. Since g′(x) has different one sided limits at x = 0, g cannot be differentiable there.
Step 2
But this g is a sufficiently good antiderivative for computing definite integrals of f, as you can see by splitting the integral in two parts: a b = a 0 + + b if a < 0 < b

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