We have to find the differentiability and the continuity of the function which we will do by considering both graphical and algebraic methods.

Now f(x) = │log│x││, as we know that log x function is continuous in its domain and hence it is also derivable, but after we introduce modulus function, there might be the formation of sharp corners as the graph’s direction changes to its mirror reflection.

The graph of the function is,

Now we observe the two red dots which are sharp end corners due to mirror image formation because of outer modulus function due to which we observe that the function is non-derivable and there are two curves due to inner modulus function which does not connect and hence the function is not continuous.

We can also prove this question by the algebraic method by reducing the function to a much simpler form, which is,

f (x) = {

{

Therefore this is the resolved function which is to be used to prove that weather this function is continuous and derivable or not.