How to find whether this function is differentiable or not at x=1?? ( because it contains signum and fractional part functions)

# Doubts from differentiablilty

For x=1 you can see, between 0 and 1 graph is x^2+x and from 1 to 2 it is x^2+x-1 so it is not continuous at 1 and thus non differentiable.

How is it x^2 + x and x^2 + x + 1??

Between 0 to 1, sgn will be 1 and {x} will be simply x. Same logic for 1 to 2

But there is GIF inside signum na… So for x between 0 to 1… GIF of x will be 0 and signum for 0 is 0… So it should be just x right??

And for 1 to 2, how did that extra 1 come from??

Oops right i was talking that gif wrongly as 1. For 1 to 2, sgn would be 1 and {x} would be x-1. Sorry for the mistake. So the graph will be non differentiable at 1 because of sharp turn- from y=x to y=x^2+x-1

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Understood… Thanks!