The set of points where

Question:

The set of points where f(x) = x – [x] not differentiable is ____________.

Solution:

Let $g(x)=x$ and $h(x)=[x]$.

Every polynomial function is differentiable for all $x \in \mathrm{R}$. So, $g(x)=x$ is differentiable for all $x \in \mathrm{R}$.

Also, the function $h(x)=[x]$ is discontinuous at all integral values of $x$ i.e. $x \in$ Z. So, $h(x)=[x]$ is not differentiable at all integral values of $x$ i.e. $x \in Z$.

Now, $f(x)=g(x)-h(x)=x-[x]$

So, the function $f(x)=x-[x]$ is differentiable for all $x \in \mathrm{R}$ except at all integral values of $x$ i.e. $x \in \mathrm{Z}$. The function $f(x)=x-[x]$ is not differentiable for all $x \in$ $R-Z$.

Thus, the set of points where f(x) = x – [x] not differentiable is R − Z.

The set of points where f(x) = x – [x] not differentiable is ___R − Z___.

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