This is another conjunction whose componentes sugggest something that has nothing to do with the actual semantic value. Contudo sounds a little bit like com tudo, with everything. However contudo is a synonym for no entanto, porém, , todavia. Words can get far away from the etymological origin what is not a big problem in general, the meaning of lots of words used nowadays has completely changed their semantic value in the course of history. However if it competes with other similar words with a different meaning, there is a problem.
O artista convenceu sua plateia, contudo não foi bem aceito pela crítica.
The artist convinced his audience, but was not well received by the critics.
Contudo, estes serviços são muitas vezes inexistentes.