Corollary of Intermediate Value Theorem

[yuching]

If I is an interval in real numbers and f: I -> real numbers is continuous on I, then f(I) (the image of f) is an interval in real numbers. (That is continuous images of an interval are intervals.

An interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. Therefore, let a, b be elements of f(I), and suppose (without loss of generality) that a < b. Then if a < q < b, we want to show that q is an element of f(I). Since a, b are elements of f(I), then there are points u, v that are elements of I such that a = f(u) and b = f(v). Consider the function x -> f(x) - q. Then...


My work is: Pick a, b elements of I such that f(a) is less than or equal to f(x) for all x element of I and f(b) is greater than or equal to f(x) for all x element of I. f(I) is on the interval [f(a), f(b)] using the intermediate value theorem.

Help please? Thank you

 

[Unco]

Hi Yuching,

Your work is a bit befuddled, but real analysis takes some time to get used to.

The question has given you a nice setup. Consider that function x -> f(x) - q they suggest, which maps from I into the real numbers, and call it g; i.e., g(x) = f(x) - q. Now, given that a < q and b > q, and f(u) = a and f(v) = b -- this is just from what the question has told us -- what can you say about g(u) and g(v)? In particular, what are the signs (negative/positive), and what does the Intermediate Value Theorem then tell you? Make sure you know, or can refer to, what exactly the statement of the theorem is.