Will smoothing splines always lead to continuous $\hat{f}$?

Question

I found these notes referencing smoothing splines in elements of mathematical learning. I'm just confused why (3) is False because I really can't think of a counter example

1. λ can be chosen by cross-validation : True
2. If λ = 0 and $x_i$ are different, smoothing splines will lead to a training error of zero: True
3. Smoothing splines will always lead to continuous $\hat{f}$ : False
4. Larger λ correspond to less flexible models: True

Definition: