Here's a question from a real analysis textbook:
Let $f:[0,1]\to\mathbb{R}$ be continuous. Assume that the image of $f$ lies in $[1,2]\cup(5,10)$ and that $f(1/2)\in [0,1]$. What can you conclude about the image of $f$?
Here are my questions on this question: Since $f$ is continuous function and undoubtedly, $[0,1]$ is an interval, shouldn'timage of $f$ be an interval?
What does it mean for a image set to lie in some set?
It has been been given that $f(1/2)\in [0,1]$ and if the image is $[1,2]\cup(5,10)$ that forces me to conclude $f(1/2)=1$, so most likely the image is not $[1,2]\cup(5,10)$.
Asserting that the image lies in $[1,2]\cup(5,10)$ means that the image of $f$ is a subset of $[1,2]\cup(5,10)$, not that it is equal to it. You are right when you deduce that $f\left(\frac12\right)=1$. Since $f$ is continuous, it follows from this fact and from the intermediate value theorem that the image of $f$ lies in $[1,2]$, since it is the largest interval in $[1,2]\cup(5,10)$ to which $1$ belongs.