Definition: A knot can be defined as an embedding $f:S^1 \to \mathbf{R}^3$, i.e. as a continuous, injective map from $S^1$ to $\mathbf{R^3}$ which realizes a homeomorphism on its image (Porter - Knots and Surfaces (1994), p46. for embedding, p52. for knot).
I fully understand the following points:
- Requiring the continuity of $f$ is natural, otherwise the image of $S^1$ could be in several pieces.
- Requiring the injectivity of $f$ is also very understandable, otherwise double points would appear.
Question: Is the continuity of the function $f^{-1}: f\left(S^1\right) \to S^1$ really necessary when defining a knot?
I tried to find examples of continuous and injective maps $f:S^1\to\mathbf{R}^3$ strange enough for a problem to arise, without success.
No. Any continuous, injective map from a compact space into a Hausdorff space is a homeomorphism on its image.
See for example this question for a proof.