I have been reading Hirsch's differential topology and now he introduces the concept of Jets between manifolds. He claims two things
If $M$ and $N$ are $C^{s+r}$ manifolds then $J^r(M,N)$ will be a $C^s$ manifold.
This I was able to see in an intuitively way but I am not quite sure how to write the map in local coordinates
For each $C^r$ map between $M$ and $N$ we can define $J^r f: M\rightarrow J^r(M,N)$ that sends $x$ to $j_x^r f$. If $M$ and $N$ are $C^{s+r}$ manifolds then $J^ r f$ is a $C^r$ map.
Now this one I can't even see intuitively why it would be true since it seems to me that when we try to write things in local coordinates there will appear terms of $f$ this is just a $C^r$ function.
Say let's look at what's happening at the local representations, we start with $x=(x_1,...,x_m)$ in $\mathbb{R}^m$ then this is sent to $\phi^{-1}(x)$ and this is sent to $[\phi^{-1}(x),f,U]_r$and this is sent to $[\phi'\circ\phi^{-1}(x),\psi\circ f\circ \phi',\phi'(U)]$ and then we want to do the taylor series of this and the $r-$derivative term will look like $D^r( \psi\circ f\circ \phi')(\phi'\circ\phi^{-1}(x))D^r(\phi'\circ \phi^{-1})(x)$, and this is where I am confused because I can't see why this term would be $C^s$, it seems to me it's only continuous since we can't differentiate $f$ anymore.
Any help or hints with this are appreciated. Thanks in advance.