# Analysis question no.2

• February 4th 2011, 06:40 AM
maximus101
Analysis question no.2
if we know that f : [a,b] ----> reals is continuous. How can we prove that the image f([a,b]) is a closed interval say [m,M] for some m,M in reals?
• February 4th 2011, 06:43 AM
HallsofIvy
A continuous function maps a compact set to a compact set.
• February 4th 2011, 08:35 AM
tonio
Quote:

Originally Posted by maximus101
if we know that f : [a,b] ----> reals is continuous. How can we prove that the image f([a,b]) is a closed interval say [m,M] for some m,M in reals?

Or, perhaps a little simpler answer than HallsofIvy's, put $f(a)=m\,,\,f(b)=M$ and now use the

intermediate value theorem for continuous functions...

Tonio