Order of continuity of an ANN approximation dependent on the activation functions used?

If I have understood this correctly, a result from Hornik et al.'s Universal Approximation of an Unknown Mapping and Its Derivatives Using Multilayer Feedforward Networks essentially states that, if we have a function $$F\in C^n$$ and a neural network $$F_{ANN}$$ that approximates $$F$$, then, if an activation function that is of class $$C^n$$ is used, then $$F_{ANN}$$ arbitrarily approximates $$F$$ and all its derivatives up to order $$n$$.

($$F$$ is said to be of class $$C^{n}$$ if its first through to $$n$$ derivatives are continuous).

If, for example, we have a neural network with 3 hidden layers, each using an activation function that is of class $$C^n$$, and an output layer that simply uses the linear (identity) activation function, does the network still approximate $$F$$ and all of its derivatives up to order $$n$$?

My thinking is that because the output layer activation is just the linear activation function, the answer is yes. But I am not sure, as the activation function used in the hidden layers (which is of class $$C^n$$) is not used on the output layer.