I am fed up by asking continuous questions.
Can I say like this?
No. One is fed up "with" something, not "by". You can say this: I am fed up with continually asking questions.
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No. One is fed up "with" something, not "by". You can say this:
I am fed up with continually asking questions.
Most often we are "fed up" with things outside our control, whereas "asking questions" is apparently under the speaker's control. This doesn't mean it's wrong, though, if the speaker feels forced or obliged to do it, but you could consider this instead:
I a