Can we substitute a linear function while integrating a function which is discontinous.

If not then what can be the complications when we get wrong answer doing that.

I have observed that many students don't check for continuity and differentiability of function while substitution. they take it for granted that it will be true. Isn't it wrong.

  • 3
    $\begingroup$ Could you give an example to illustrate your point? $\endgroup$
    – Henry
    Jan 27 '19 at 12:22
  • $\begingroup$ I have read in Sir IA Maron calculus book that 3 points must be considered while substitution. I didn't get the third point as in every function we were just substituting. Even in discontnous function. Like in int(tanx,0,2π) $\endgroup$
    – Amir RAZA
    Jan 27 '19 at 12:44
  • $\begingroup$ It's not clear to me what you're asking, but maybe it's something like this: math.stackexchange.com/questions/1356523/…? $\endgroup$ Jan 27 '19 at 13:06

The only complications to an integration by substitution come from irregularities in the function being substituted. What's irregular here? Any discontinuities - that's pretty much automatically fatal. Zero derivative somewhere - because we might have to divide by it. Not being monotone - because then the interval folds back onto itself. Not so bad if we have the derivative of the substitution there, but serious trouble if we don't. An inverse we can't calculate cleanly - this doesn't really impact the theory, but it'll likely ruin our chance of getting a nice formula for the answer.

If that's a linear (affine) function we're substituting, none of these irregularities apply. Affine substitutions always work. They're worth using even if only to gain some minor convenience.

As for the function we're integrating having discontinuities - that's not going to matter. They'll still be there after the substitution, just the same.

  • $\begingroup$ An example will be deeply appreciated $\endgroup$
    – Amir RAZA
    Jan 27 '19 at 12:42

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