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to continue
To continue with an activity or action that has been started, often after an interruption.
Nach einer kurzen Pause machten wir mit der Gartenarbeit weiter.
After a short break, we continued with the gardening.
'Weitermachen' (to continue/carry on) is a separable verb. The prefix 'weiter' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Wir machen nach der Pause weiter.
We continue after the break.
The verb is logically made up of 'weiter' (in the sense of 'forward' or 'continuing') and 'machen' (to do/make). So it literally means to continue an action.
Trotz des Regens machten die Bauarbeiter einfach weiter.
Despite the rain, the construction workers simply carried on.
If you want to say exactly what you are continuing with, use the preposition 'mit' followed by the dative. The structure is: mit etwas weitermachen (to continue with something).
Lass uns mit dem nächsten Thema weitermachen.
Let's continue with the next topic.
'Weitermachen' (to carry on) is very often used as a command or encouragement to motivate someone not to give up. It often appears in the imperative.
Gib nicht auf, mach einfach weiter!
Don't give up, just carry on!
ball
'am Ball bleiben' (to stick with it) and 'weitermachen' (to carry on) can both mean that you do not give up. 'weitermachen' is a normal verb and says directly that you continue an activity, even when it is difficult. 'am Ball bleiben' is a fixed expression. It sounds more figurative and emphasizes that you stay active and engaged in something. 'weitermachen' fits in many different situations. 'am Ball bleiben' fits especially well when talking about a goal, a project, or regular practice.
Nach der schlechten Note wollte Tom trotzdem weitermachen.
After the bad grade, Tom still wanted to carry on.
Beim Deutschlernen musst du am Ball bleiben, auch wenn die Grammatik schwer ist.
When learning German, you have to stick with it, even when the grammar is difficult.