Antiderivative of discontinuous functionI am having confusion regarding anti-derivative of a function. f ( x ) = { − x 2 2 + 4 x ≤ 0 − x 2 2 + 2 x &gt; 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 ]?

Question

Antiderivative of discontinuous functionI am having confusion regarding anti-derivative of a function.  f  (  x  )  =      {                            −                                    x              2                        2                    +          4                          x          ≤          0                                                                −                                                          x              2                        2                    +          2                          x          &amp;gt;          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  ]?

rynosluv101swv2s · Accepted Answer

Step 1No. 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 2But 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  &amp;lt;  0  &amp;lt;  b

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

Answered question

Answer & Explanation

New Questions in Calculus 1