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Addition of non differentible and differentiable function is always non differentible .

But is the subtraction of non differentible and differentiable function is also always non differentible ?.

Here is one example f(x)=sin(|x-1|)-|x-1| check differentiability at x=1 SOLUTION sin(|x-1|) in differentiable at x=1 but |x-1| is not so f(x) should be non differentiable at x=1 but if we check though the formula ( RHD AND LHD ) f(x) is differentiable

Raunii
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Non-differentiable minus differentiable is non-differentiable (since if $f$ is differentiable then so is $-f$, so you can just use "sum of non-differentiable and differentiable is non-differentiable" with $-f$ instead of $f$).

However, this doesn't help you in your example, since you assert that $\sin(|x-1|)$ is differentiable at $x=1$ when in fact it is not. Your example is the sum of two non-differentiable functions, so it may or may not be differentiable. As you point out, it's actually differentiable.

Patrick Stevens
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  • Can you explain me how sin(|x-1|) is not differentiable at x= 1 – Raunii Apr 08 '17 at 07:48
  • Why should it be differentiable? Most functions are not differentiable. It's the composition of a differentiable and non-differentiable function, so there's no reason a priori to believe it should be differentiable. But if you want a proof: for $x<1$ sufficiently near to $1$ it's $\sin(1-x)$, and for positive $x$ sufficiently near to $1$ it's $\sin(x-1)$. The derivative of the first function is $-\cos(1-x) = -\cos(x-1)$; the second function, $\cos(x-1)$. So near $1$, the left-hand derivative is near $-1$; the right-hand derivative is near $1$. – Patrick Stevens Apr 08 '17 at 07:55
  • But sir from the graph of sin(|x-1|) I observed that it is differentiable at x=1 – Raunii Apr 08 '17 at 08:09
  • @Raunii What's its derivative? – Patrick Stevens Apr 08 '17 at 08:09
  • Ok I realized that it is not differentiable at x=1 by taking the derivative . I think I am not drawing the graph correctly , can you help me with the graph of f(x)=sin(|x-1|) – Raunii Apr 08 '17 at 08:15