The motion results in a static line with no gaps or jumps. The static line has no directionality. The mental picture here is one of preserving closeness near a point. For, if it were not, then talk about x approaching a would not capture the conception we need. In this conception, a line or a continuum is a fundamental object in its own right.
Points are simply locations on lines.
When we formulate the final Cauchy-Weierstrass definition, however, by making precise the notion of a limit, we abandon the dynamic view, based on the idea of a gapless real continuum, and replace it by an entirely static conception that speaks about the existence of real numbers having certain properties. The conception of a line that underlies this definition is that a line is a set of points. The points are now the fundamental objects, not the line. This, of course, is a highly abstract conception of a line that was only introduced in the late nineteenth century, and then only in response to difficulties encountered dealing with some pathological examples of functions. Rather, we are changing the conception of continuity in almost every respect.