Does the Box-Counting Dimension always exist after applying a Lipschitz mapping?

Question

When you have a bi-Lipschitz mapping and the Box-Counting Dimension of $A$ is defined, then the Box-Counting Dimension of $f(A)$ ist also defined (and the same as the B.-Dim. of $A$).
We can also conclude that the upper and lower B.-Dim. of $f(A)$ are bounded by the upper/lower B.Dim. of $A$ but we don't know if the upper and the lower B.Dim. are equal.
Is there an example where the Box-Counting Dimension of $f(A)$ is not well defined with $f$ a Lipschitz mapping?