If A denotes the set of continuous functions and B denotes the set of differentiable functions, then which of the following depicts the correct relation between set A and B?
Relationship Between Continuous and Differentiable Functions
The relationship between continuous and differentiable functions is fundamental in calculus. Every function that is differentiable at a point is also continuous at that point. However, the converse is not necessarily true; a function can be continuous at a point but not differentiable there (e.g., \( f(x) = |x| \) at \( x = 0 \)).
Let \( A \) be the set of continuous functions and \( B \) be the set of differentiable functions. Since every differentiable function is continuous, the set \( B \) is a subset of the set \( A \). This can be represented as \( B \subseteq A \).
Analysis of Venn Diagrams:
Diagram shows that \( A \) is a subset of \( B \), which is incorrect.
Diagram (B) shows that \( B \) is a subset of \( A \), which is correct.
Diagram (C) shows an overlap between \( A \) and \( B \), implying some functions are only continuous, some only differentiable, and some both. While there are functions that are continuous but not differentiable, there are no differentiable functions that are not continuous.
Diagram (D) shows \( A \) and \( B \) as disjoint sets, which is incorrect as all differentiable functions are continuous.
Therefore, the correct Venn diagram is the one where the circle representing the set of differentiable functions (\( B \)) is entirely contained within the circle representing the set of continuous functions (\( A \)).
