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is the asian option payoff. Which is also clearly pathwise continuous. How can i mathematically show that it is continuous?

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The question does not make much sense. You can prove however that $$T \mapsto \left( \frac{1}{T-t}\int_t^T S_u du - K\right)_+$$ is pathwise continuous, which is just a composition of continuous functions, if you work on the set where $t \mapsto S_t(\omega)$ is continuous.

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