Let $a,b \in \Bbb{R}$ such that $a&lt;b$ Then Prove that $\:\exists\:$ a continuous function $f:(a,b)\to [a,b]$ such that $f$ is Bijective. I am not sure how to prove formally, But i tested the claim for $f(x)=x$. It is of course injective but not surjective since codomain is different from range. EDIT: As per the link given in the comment section, this is what i understood. First let us assume $f:(a,b)\to [a,b]$ is onto. So $\exists$ $c,d \in (a,b)$ such that $f(c)=a,f(d)=b$. Without loss of generality let $c&lt;d$. Now Since $f$ is continuous, by IVT we have: $$f\left([c,d]\right)=[a,b]$$ But $(a,b)-[c,d]\ne \phi$ So $f$ cannot be injective. Hence if $f$ is Surjective, $f$ cannot be injective. This implies that: If $f$ is Injective, $f$ cannot be surjective. The best example for $f$ to be continuous and injective is $f(x)=x$. But I am unable to find example for $f$ to be continuous and surjective.
Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
Regression with only categorical variables How do I block unwanted callers without jailbreaking? How can a lens with a single focal length focus on more than one plane? Fantasy book where a young prince becomes a mage When I was younger (11-12 years old, so the early 2000s) I once read a story where a young boy, who may have been either a young prince or a squire of some sort, is recruited by a wizard and taken to a city to train as a mage.  I seem to remember him living in dorms with the other children at this school, until he gets into difficulty with another of the boys in the dorm. He is kicked out of the dorms and ends up living above a blacksmith shop (I think).  I remember it being the first book in a series of books of which there were about 3 out at the time. I think one of the stories was about the boy going to a tower to do something or other.  Does anyone have any idea what this book series is? I haven't been able to find it for years now but it was a story I really loved. 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