Test Function/Examples

Examples of Test Functions

Exponential of $\dfrac 1 {x^2 - 1}$

The graph of the test function. It is smooth everywhere. Outside of its support denoted by dashed lines the function is identically zero. At boundary points it connects smoothly.

Let $\phi : \R \to \R$ be a real function with support on $x \in \closedint {-1} 1$ such that:

$\map \phi x = \begin {cases}

\map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$

Then $\phi$ is a test function.