Proving Continuity and Inverting a Piecewise Function

The concept of continuity in mathematical functions is fundamental, particularly when one desires to apply techniques like differentiation or integration. Continuous functions have a smooth and unbroken graph, whereas discontinuous functions may exhibit sharp breaks or jumps. In this article, we will explore a specific function and analyze its continuity. We will also discuss how to find its inverse and ensure the continuity of the inverse function.

Introduction to the Function

Consider the function ( f(x) ) defined as:

[ f(x) begin{cases} e^{-|x|} text{if } x eq 0 0 text{if } x 0 end{cases} ]

At first glance, it might seem that this function is not continuous because of the specific definition at ( x 0 ). However, we need to rigorously prove this, and if it is not continuous, we will determine if it has an invertible inverse and under what conditions.

Analysis of Continuity at ( x 0 )

To prove the continuity of ( f(x) ) at ( x 0 ), we need to evaluate the limit of ( f(x) ) as ( x ) approaches 0 and see if it equals ( f(0) ).

For ( x > 0 ), the function becomes ( f(x) e^{-x} ).

For ( x

Let's evaluate the limits from both directions:

Limit from the Right (( x to 0^ ))

[ lim_{x to 0^ } f(x) lim_{x to 0^ } e^{-x} e^0 1 ]

Limit from the Left (( x to 0^- ))

[ lim_{x to 0^-} f(x) lim_{x to 0^-} e^{x} e^0 1 ]

Since both the right-hand limit and the left-hand limit are equal to 1, and ( f(0) 0 ), the function is not continuous at ( x 0 ). This can be seen from the graph, which has a jump discontinuity at ( x 0 ).

Finding the Inverse Function

Despite the discontinuity at ( x 0 ), the function ( f(x) ) is still invertible. We can define a piecewise inverse function as follows:

Defining the Inverse for ( x eq 0 )

For ( x > 0 ), we have ( y e^{-x} ) and solving for ( x ) gives us:

[ x -ln(y) ]

For ( x

[ x ln(y) ]

Thus, the inverse function ( f^{-1}(y) ) can be written as:

[ f^{-1}(y) begin{cases} -ln(y) text{if } y > 0 0 text{if } y 0 ln(y) text{if } y

Ensuring Continuity of the Inverse Function

Now, we need to ensure that the inverse function ( f^{-1}(y) ) is continuous. Let's analyze the behavior of ( f^{-1}(y) ) as ( y ) approaches 0 from both sides.

Continuity from the Right (( y to 0^ ))

[ lim_{y to 0^ } f^{-1}(y) lim_{y to 0^ } -ln(y) infty ]

From the right, as ( y ) approaches 0, ( f^{-1}(y) ) approaches positive infinity.

Continuity from the Left (( y to 0^- ))

[ lim_{y to 0^-} f^{-1}(y) lim_{y to 0^-} ln(y) -infty ]

From the left, as ( y ) approaches 0, ( f^{-1}(y) ) approaches negative infinity.

Since the limits do not exist, the inverse function ( f^{-1}(y) ) is not continuous at ( y 0 ), but it is still a valid inverse function.

Final Thoughts

In conclusion, we have proved that the function ( f(x) begin{cases} e^{-|x|} text{if } x eq 0 0 text{if } x 0 end{cases} ) is not continuous at ( x 0 ). However, it still has an invertible inverse function. The piecewise inverse function ( f^{-1}(x) ) is valid but not continuous at ( x 0 ).