First of all you have to check the continuity of the function at each joint. f.i, if the function f(x) is defined as:
2*x-1 for x<=3
x^2-1 for x>3
first check the continuity in 3 that means the limit ot the first function when x-->3 has the same value of the
limit (when x-->3) of the second function. In our example if x=3 we have
5
8
the the function is not continuos hence not differentiable.
Let's try with:
2*x-1 for x<=3
x^2-4 for x>3
now it's continuos. We have now to check the differentiability. Calculate the first derivative:
2 for x<=3
2*x for x>3
again check both the subfunctions have the same limit for x-->3. We have:
2
6
the this function is continuous but not differentiable.
If, instead we have
6*x-13 for x<=3
x^2-4 for x>3
the limit for x-->3 is 5 in both cases. The first derivative is:
6
2*x
then the limit of the derivative for x-->3 is 6 in both cases. The function is differentiable.
When I applied the limit to the first of the two subfunction on each f(x) I was coming from 0 towards 3, because that subfunction is defined for x<=3. So I'm moving from left to right that means I'm using all values a little bit less than 3 then I'll write lim x-->3-. When instead I applied the same limit to the second subfunction, I was moving from right to left, that means I was using all values a little bit greater than 3 then I'll write lim x-->3+
View attachment 109868