Final answer:
To prove that f is a homeomorphism, we need to show that it is a continuous, one-to-one, and onto function.
Step-by-step explanation:
To prove that f is a homeomorphism, we need to show that it is a continuous, one-to-one, and onto function. Since M is compact and f is continuous, we know that the image of M under f is also compact.
Next, we can show that f is one-to-one by assuming that there are two distinct points in M that map to the same point in N. Since f is continuous and one-to-one, this cannot happen.
Finally, we can show that f is onto by assuming that there exists a point in N that does not have a pre-image in M. Again, by continuity, this cannot happen. Therefore, f is a homeomorphism.
Since M is compact and f is continuous, we know that the image of M under f is also compact. Next, we can show that f is one-to-one by assuming that there are two distinct points in M that map to the same point in N.
Finally, we can show that f is onto by assuming that there exists a point in N that does not have a pre-image in M. Therefore, f is a homeomorphism.