In Tu's book on Manifolds, specifically in chapter 2, paragraph 6 "Smooth Maps on a Manifold", I came across the definition of the smooth function at a point $p$ in (a manifold) $M$, which goes like
Let $M$ be a smooth manifold of dimension $n$. A function $f:M\rightarrow\mathbb{R}$ is said to be $C^{\infty}$ or smooth at a pont $p$ in $M$ if there is a chart $(U,\phi)$ about $p$ in $M$ such that $f\circ\phi^{-1}$, a function defined on the open subset $\phi(U)$ of $\mathbb{R}^n$, is $C^{\infty}$ at $\phi(p)$ . The function $f$ is said to be $C^{\infty}$ on $M$ if it is $C^{\infty}$ at every point of $M$.
Then, in the next page there exists the following paragraph
In Definition 6.1, $f:M\rightarrow\mathbb{R}$ is not assumed to be continuous. However, if $f$ is $C^{\infty}$ at $p \in M$, then $f \circ \phi^{−1} : \phi (U)\rightarrow\mathbb{R}$, being a $C^{\infty}$ function at the point $\phi(p)$ in an open subset of $\mathbb{R}^n$, is continuous at $\phi(p)$. As a composite of continuous functions, $f = ( f \circ \phi^{−1}) \circ\phi$ is continuous at $p$. Since we are interested only in functions that are smooth on an open set, there is no loss of generality in assuming at the outset that $f$ is continuous
My question is the following:
How can the function $f$ be $C^{\infty}$, but not continuous? Isn't continuity at the very core of the definition of $C^{\infty}$? Can someone elaborate on that last paragraph?
Tu's whole point is that a function cannot be $C^\infty$ without being continuous, even though continuity is not part of the definition of $C^\infty$.