Please disregard the first eight lines of the solution below (which I have provided for completeness; the referenced theorems simply state that continuous functions between metric spaces preserve compactness and connectedness).

My question concerns only lines 5 and 3 from the bottom. In each of these cases, while $t$ certainly equals $f(x,y)$ for some $(x,y) \in \mathbb R^2$, the $x^2+y^2\in[1,2]$ so that $(x,y) \notin K$-- i.e., contrary to the author's assertion, we have not shown that $t\in f(K)$, right?

> 
> Daniel has pointed out in the comments that $f(K)$ is actually $[-1,\frac 54]$.
> 
> 
> How to show that $t\in (1,\frac 54]\implies t\in f(K)$? And that $t\in f(K)\implies t\in [-1,\frac 54]$?
> 
> 
> 

![enter image description here](https://i.stack.imgur.com/GARXN.jpg)

P.S. This exercise is from Marsden's Elementary Classical Analysis.