# Test function

## Homework Statement

Investigate whether $$\psi$$(x) = $$\phi$$(c(x - $$\eta$$)) & $$\psi$$(x) = $$\phi$$(x2) are test functions.

## The Attempt at a Solution

The first function is smooth but has no contact support as it is only 0 at x = $$\eta$$ so this is not a test function.
The second function is smooth but is not 0 at any interval, so this is not a test function.

## The Attempt at a Solution

But you did not say what are the assumptions about $$\phi$$. Without knowing them one can't answer these questions.

Sorry, assume that $$\phi$$(x) is a test function on (-$$\infty$$,$$\infty$$)

They are both test functions I think.

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You see, there are many kinds of test function spaces. You probably have learned about just one. To see that some function is a test function in the sense you know it, you need to check the precise definition of your test function space. Which conditions a given function must satisfy to be a test function? Differentiable? How many times? Compact support? Or vanishing sufficiently fast at infinity?