What problem you could face when you convert some removable discontinuous functions to continuous functions…??

E.g. let f(x):=(x^2) sin(1/(x^2))..

To make it continuous we define the function at zero to be zero.

In that case we get also that the function is differentiable everywhere…

But you can easily see that the derivative function gets discontinuous at point zero…

You actually see that we have the derivative function as : 2xsin(1/(x^2))-(2/x)cos(1/(x^2))….

Limit of that f'(x) does not exist although f'(0)=0 very weiredly not as usual functions do…. Even we have not defined f'(0) to be zero it’s got the value automatically… Not actually as we see… We actually defined f(0)=0… In that case although we thought we have not mess up anything but still a huge mess up for the weiredness of the f'(x) was generated then!!!!

Just watch the plotting of the both f(x) & f'(x) below in Desmos( graphing calculator)…

Graph of f(x)

Graph of f'(x)