Demonstrating Continuity and Non-Differentiability at ( x 0 ) for ( f(x) x|x| )
Demonstrating Continuity and Non-Differentiability at ( x 0 ) for ( f(x) x|x| )
Understanding the behavior of functions at specific points is fundamental in calculus. This article will guide you through a detailed analysis of the function ( f(x) x|x| ) to showcase its continuity and non-differentiability at ( x 0 ).
The Function and Its Piecewise Representation
First, let's define the given function ( f(x) x|x| ) in a piecewise form based on the definition of the absolute value function. The absolute value function ( |x| ) behaves differently on either side of ( x 0 ).
Definition:
[ f(x) begin{cases} 2x text{if } x geq 0 0 text{if } x 0 end{cases} ]
Continuity at ( x 0 )
For a function to be continuous at a point ( x a ), it must satisfy three conditions:
Limit Existence: The limit as ( x ) approaches ( a ) must exist. Function Value Existence: The function value at ( x a ) must exist. Equality of Limit and Function Value: The limit as ( x ) approaches ( a ) must equal the function value at ( x a ).Calculations
Let's verify these conditions for ( x 0 ).
Step 1: Verify ( f(0) )
When ( x 0 ):
[ f(0) 0 |0| 0 ]
Step 2: Left-Hand Limit as ( x ) Approaches 0
For ( x 0 ), the function simplifies to:
[ f(x) 0 ]
Thus, the left-hand limit is:
[ lim_{x to 0^-} f(x) lim_{x to 0^-} 0 0 ]
Step 3: Right-Hand Limit as ( x ) Approaches 0
For ( x geq 0 ), the function simplifies to:
[ f(x) 2x ]
Thus, the right-hand limit is:
[ lim_{x to 0^ } f(x) lim_{x to 0^ } 2x 2(0) 0 ]
Conclusion
Since both the left-hand and right-hand limits are equal to ( f(0) ), the function is continuous at ( x 0 ).
Non-Differentiability at ( x 0 )
To check differentiability at a point, we need to find the left-hand and right-hand derivatives and see if they are equal.
Left-Hand Derivative at ( x 0 )
For ( x 0 ):
[ f(x) 0 ]
The derivative is:
[ f'(x) 0 ]
Hence, the left-hand derivative at ( x 0 ) is:
[ f_0^- 0 ]
Right-Hand Derivative at ( x 0 )
For ( x geq 0 ):
[ f(x) 2x ]
The derivative is:
[ f'(x) 2 ]
Hence, the right-hand derivative at ( x 0 ) is:
[ f_0^ 2 ]
Conclusion
Since ( f_0^- ) and ( f_0^ ) are not equal, the function ( f(x) x|x| ) is not differentiable at ( x 0 ).
Conclusion
We have demonstrated that the function ( f(x) x|x| ) is continuous at ( x 0 ) but not differentiable at the same point. Understanding these distinctions helps in grasping the nuances of calculus and function behavior.