Vector fields that make up a local frame are continuous

Question

In Lee's book Introduction to Smooth Manifolds, on the section on orientations, he notes that

Recall that by definition the vector fields that make up a local frame are continuous

which I did not fully understand. If we are given a set of vector fields $X_1, \ldots, X_n$, then them being a local frame on some open set $U$ of a manifold $M$ means that for each $p \in U$, $\{X_i\}$ forms a basis for $T_pM$. How is continuity implied from this definition?

In fact, in Tu's book Introduction to Manifolds he explicitly mentions that the vector fields may be discontinuous:

A frame on an open set $U \subset M$ is an $n$-tuple $(X_1, \ldots, X_n)$ of possibly discontinuous vector fields on $U$ such that at every point $p \in U$, the $n$-tuple $(X_{1,p}, \ldots, X_{n,p})$ of vectors is an ordered basis for the tangent space $T_pM$.

Answer 1

The vector field defined on Chapter 8 (page 174) is a continuous map from $M$ to $TM$.

If $M$ is a smooth manifold with or without boundary, a vector field on $M$ is a section of the map $\pi$: $TM\rightarrow M$. More concretely, a vector field is a continuous map $X$: $M\rightarrow TM$ with the property that $\pi\circ X ={\rm Id}_M$

And therefore the vector fields of the local frame defined on page 178 are all continuous.

Also, be careful, the local frame is defined on an open set, otherwise the continuity of the pointwise orientation may still fail. (Mobius band can help you think of this counterexample.)

Here are some points of definitions related to your question in Lee's book:

Vector field ----- continuous but may not smooth (page 174)

Rough vector field ----- may not continuous and may not smooth (page 175)

Smooth vector field ----- continuous and smooth (page 175)

Local frame ----- continuous but may not smooth (page 178)

Smooth frame ----- continuous and smooth (page 178)

Lee didn't define "continuous local frames" you said, since local frames are continuous in his book.