Show from Eq. 1.1 that the below function is smooth at t = 1 and at t = 2. Is it smooth at any 1 < t < 2?

x(t) =

1.0 + 2.0 t 0 ≤ t ≤ 1

3 + 4(t − 1) 1 ≤ t ≤ 2

7 + 3(t − 2) 2 ≤ t

for equation 1.1 my book gives me:

lim

dt→0 [x(t + dt) − x(t) ]/dt= 0

This problem asks me to show that the motion is smooth, but to me it seems that it would not be smooth at points 1 and 2. the way i understand it, for a motion to be smooth all of the derivatives of x(t) must exist, but if i take the derivatives:

2 0 ≤ t ≤ 1

4 1 ≤ t ≤ 2

3 2 ≤ t

the derivatives at points 1 and 2 conflict, so i dont understand how this meets the requirements of a smooth function. i must be missing something obvious.

Also i was confused by the formula 1.1 in the book, it tell me to use this formula to show the motion is smooth, but then next to the formula in the book it states the formula as being one used to show a function is continuous.

Thank you in advance to anyone who responds.