Let $ \pi_{ 1 } \colon E_{ 1 } \to M $ and $ \pi_{ 2 } \colon E_{ 2 } \to M $ be smooth vector bundles (of finite rank) over a smooth manifold $ M $, and consider a map $ T \colon \Gamma ( E_{ 1 } ) \to \Gamma ( E_{ 2 } ) $ from the smooth sections of one to the smooth sections of the other. If $ T $ is $ \mathrm{ C }^{ \infty }( M ) $-linear, then $ ( T \sigma )_{ p } $ depends only on $ \sigma_{ p } $.
However, this need not be true if $ T $ is only $ \mathbb{ R } $-linear. For example, for a Koszul connection $ \nabla $ in a smooth vector bundle $ \pi \colon E \to M $, $ ( \nabla_{ X } \sigma )_{ p } $ depends not only on $ \sigma_{ p } $ but on the values of $ \sigma $ in an arbitrarily small neighborhood of $ p $.
Does this "locally determined" property hold for general $ T $ defined as above but only required to be $ \mathbb{ R } $-linear? That is, is it true that if $ T \colon \Gamma ( E_{ 1 } ) \to \Gamma ( E_{ 2 } ) $ is $ \mathbb{ R } $-linear, then does $ ( T \sigma )_{ p } $ depend only on the values of $ \sigma $ in an arbitrarily small neighborhood of $ p $?
No. Take for instance $M = \mathbb{R}$, $E = E_1 = E_2 = \mathbb{R} \times \mathbb{R}$ and $\pi_1 = \pi_2$ be the projection onto the first variable. Note that if $\sigma \in \Gamma(E)$, then $\sigma(x) = (x,f(x))$, for some $f \in C^\infty(M)$. Define $T: \Gamma(E) \to \Gamma(E)$ by $$ T(\sigma)(x) = \left(x,\int_0^xf(s)ds\right). $$