Is the word continuar the same in Brazilian and European Portuguese?

No.

While the meaning, spelling, and conjugation of "continuar" are identical in both dialects, there is a significant difference in grammatical usage. In Brazilian Portuguese, "continuar" is typically followed by the gerund (the -ando, -endo, -indo form) to describe an ongoing action. In Continental Portuguese, "continuar" is almost exclusively followed by the preposition "a" and the infinitive of the verb.

Brazilian Portuguese Examples:

1. Eu vou continuar trabalhando até tarde. (I am going to continue working until late.)
2. Ela continua falando muito. (She keeps talking a lot.)
3. Nós continuamos estudando para a prova. (We are continuing to study for the test.)
4. Eles continuaram correndo na chuva. (They continued running in the rain.)
5. Você precisa continuar treinando. (You need to continue training.)

Continental Portuguese Examples:

1. Eu vou continuar a trabalhar até tarde. (I am going to continue working until late.)
2. Ela continua a falar muito. (She keeps talking a lot.)
3. Nós continuamos a estudar para o teste. (We are continuing to study for the test.)
4. Eles continuaram a correr na chuva. (They continued running in the rain.)
5. Tu precisas de continuar a treinar. (You need to continue training.)