Help with analysis?

Question

I'm having trouble with the following proof:

If the domain of a continuous function is an interval, show that the image is an interval. Give examples where the image is an open interval.

Anyone able to show me that way? Thanks

Answer 1

y = sec (x) is continuous over the open interval (pi/2, 3pi/2)

The image of y = sec (x) reflected over the x-axis is also continuous over the open interval (pi/2, 3pi/2).

It is clear that the image and preimage have the same interval if the image is a reflection of the preimage across theh x-axis or any horizontal line. If the image is an expansion of the preimage. or a rotation of the preimage, or a slide translation, or a combination of these methods, then I can see why you are having trouble.