Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $f(x)= |x|$ and $g(x) = -|x|$, show that $(f+g)(x)$ is differentiable at $0$ while neither $f$ nor $g$ is.

I understand that neither $f$ or $g$ is not differentiable at $0$ because there is a cusp point at the origin. What I don't understand is that if I added both functions together wouldn't it just equal $0$.

share|improve this question
    
why do you say that $x=0$ is a cusp point of $f?$ To my knowledge, $x=0$ is just an angular point of $f,$ according to Giaquinta's book of Mathematical Analysis, Vol I, Functions of one variable, and see Page 98 of that book. And I am very happy if you give your definition of cusp point –  azc 2 days ago

2 Answers 2

up vote 2 down vote accepted

$(f + g)(x) = |x| - |x| = 0$ which is just a constant function like $h(x) = 5$. Any constant function is differentiable, and differentiates to 0.

share|improve this answer
    
Oh I assumed that a horizontal line was not differentiable thanks –  EagleTamer Jan 13 at 0:46
    
Differentiation gives the rate of change of a function, if the function is constant everywhere then it doesn't change, hence the derivative is 0. –  Daniel Jan 13 at 0:47

The sum is in fact equal to the constant function with value zero and therefore it is differentiable..

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.