Don't worry, that's normal. My #1 always has cold hands n feet before he became more mobile (aft 6mths). Feel your baby's torso/body. If his torso is warm, your baby is ok! This is because his circulatory system is still developing, blood will go to the more important parts first :) His hands n feet will be warmer when he becomes more mobile n circulation is even out. If you are still concerned, there is no harm putting on a pair of little socks for him!
It's common. At 5 months my LO has cold and clammy feet too. Babies do have immature vervous systems and it can take a while before they are able to control their temperature fully. Many babies balance out a bit when they get more mobile, around 9 months they seem to regulate temperature better.
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Yes, it's normal. You may want to apply some oil such as Minyak Telon on his feet.