Are graphs of continuous real-valued functions locally affine varieties?*

Question

More specifically, let $f : \mathbb{R}^n \to \mathbb{R},$ where $n$ is a natural number. Can I use Taylor polynomials to show that if $f$ is continuous (or locally continuous in some neighbourhood) with the Euclidean metric, then its graph is locally an affine variety in $\mathbb{R}^{n+1}$, if the remainder term is identically zero?

Answer 1

This is a community wiki answer recording the solution from the comments, in order to remove the question from the unanswered queue:

A few comments: a locally continuous function is just a continuous function; continuous functions don't have Taylor expansions; and if the remainder term is zero in a Taylor expansion this means the function is just a polynomial. – Captain Lama