For a continuous function, a tangent touches the curve at a single point. By definition, a tangent to a curve at a point has the same slope as the curve at that point. If the tangent touches the curve at two distinct points, such as at $x = 1$ and $x = 1.0000001$, it would no longer be considered a tangent, but a secant line.

Thus, for a continuous function, the tangent cannot touch the curve at both $x = 1$ and $x = 1.0000001$ simultaneously. The tangent would only touch at a single point, typically where the derivative (the slope of the function) at that point matches the slope of the tangent.

Would you like more details on tangents or secant lines?

Here are 5 related questions for deeper understanding:

1. What is the geometric difference between a tangent and a secant line?
2. How does the derivative relate to the slope of a tangent?
3. Can a function have multiple tangents at a single point?
4. How do we find the tangent line to a curve at a given point?
5. What is the role of continuity in the existence of a tangent?

Tip: When dealing with tangents, checking the derivative at the point of interest is key to understanding its behavior.