Exercise · ZW-SMOOTH-FUNCTION-001 · ★☆☆ Basic
What does it mean for \(f\) to be of class \(C^{\infty}\)?
Choose one option.
Show answer
- B. It has derivatives of all orders in local coordinates
Explanation
The definition of a smooth function requires derivatives of all orders in local coordinates. The other options describe different properties.
Review the concept: Smooth function
A smooth function is a function between spaces with differentiable structure that, written in local coordinates, has derivatives of all orders.
\[f\in C^{\infty}(M,N)\]
More practice problems: smooth function
- ZW-SMOOTH-FUNCTION-002 Which of these functions from ℝ to ℝ is smooth?
- ZW-SMOOTH-FUNCTION-003 Which symbol denotes the set of smooth functions from M to N?
- ZW-SMOOTH-FUNCTION-004 What class is f(x)=x^5?
- ZW-SMOOTH-FUNCTION-005 Let f(x)=x|x|. What class is f?
- ZW-SMOOTH-FUNCTION-006 Let f(x)=x^2 if x≥ 0 and f(x)=0 if x<0. Is f smooth?
- ZW-SMOOTH-FUNCTION-007 If f and g from ℝ to ℝ are smooth, what can be said about g∘ f?
- ZW-SMOOTH-FUNCTION-008 Let f(x)=x^2sin (1/x) for xne 0 and f(0)=0. Which statement is correct?
- ZW-SMOOTH-FUNCTION-009 What is a diffeomorphism between two smooth manifolds?