Exercise · ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-008 · ★★☆ Intermediate
The image of \(f:\mathbb R\to\mathbb R\), \(f(x)=\dfrac{1}{1+x^2}\), is:
Choose one option.
Show answer
- B. \((0,1]\)
Explanation
It is an interval (image of a connected set). The maximum \(f(0)=1\) is attained, and \(f(x)>0\) with limit \(0\) at infinity, which is not attained: \((0,1]\).
Review the concept: Continuous image of a connected space
If \(f:X\to Y\) is a continuous function between topological spaces and \(X\) is connected, then the image \(f(X)\) is a connected subspace of \(Y\).
\[X\text{ connected},\ f\text{ continuous}\ \Longrightarrow\ f(X)\text{ connected}\]
More practice problems: continuous image of a connected space
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-001 Let f:[-1,2]→mathbb R, f(x)=x^2. The image f([-1,2]) is:
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-002 Which function f:[-1,1]→mathbb R cannot be continuous, because its image is not connected?
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-003 If X is connected and f:X→ Y is continuous, what can be guaranteed about f(X)?
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-004 Let X=\1,2,3\ with open sets ∅,\1\,\1,2\,X and Y=\a,b\ with open sets ∅,\a\,Y. How many continuous…
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-005 For all these topologies on Y=\a,b,c\, except one, there is a continuous surjection from some conne…
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-006 Which is a counterexample to «if f is continuous and f(X) is connected, then X is connected»?
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-007 Let f:[a,b]→mathbb R be continuous with f(a)=-2 and f(b)=3. Which value must f attain?
- ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-009 Onto which of these subspaces of mathbb R is there no continuous surjection from mathbb R?
← ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-007 ZW-CONTINUOUS-IMAGE-OF-CONNECTED-SPACE-009 →