The manual of mountpoint says that it:
checks whether the given directory or file is mentioned in the /proc/self/mountinfo file.
The manual of mount says that:
The listing mode is maintained for backward compatibility only. For
  more robust and customizable output use findmnt(8), especially in your
  scripts.
So the correct command to use is findmnt, which is itself part of the util-linux package and, according to the manual:
is able to search in /etc/fstab, /etc/mtab or /proc/self/mountinfo
So it actually searches more things than mountpoint. It also provides the convenient option:
-M, --mountpoint path
Explicitly define the mountpoint file or directory. See also --target.
In summary, to check whether a directory is mounted with bash, you can use:
if [[ $(findmnt -M "$FOLDER") ]]; then
    echo "Mounted"
else
    echo "Not mounted"
fi
Example:
mkdir -p /tmp/foo/{a,b}
cd /tmp/foo
sudo mount -o bind a b
touch a/file
ls b/ # should show file
rm -f b/file
ls a/ # should show nothing
[[ $(findmnt -M b) ]] && echo "Mounted"
sudo umount b
[[ $(findmnt -M b) ]] || echo "Unmounted"