Strongly meager sets and their uniformly continuous images

We prove the following theorems:

(1) Suppose that $f:2^\omega\to 2^\omega$ is a continuous function and $X$ is a Sierpinski set. Then

(A)

for any strongly measure zero set $Y$, the image $f[X+Y]$ is an $s_0$-set,

(B)

$f[X]$ is a perfectly meager set in the transitive sense.

(2) Every strongly meager set is completely Ramsey null.