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Let \(H : \mathbb{R} \to \mathbb{R}\) defined by \(H(x) = 0\) for \(x \le 0\), \(H(x) = 1\) for \(x > 0\) (Heaviside). Show that \(H\) is left-continuous at \(0\) but not right-continuous, and conclude on continuity at \(0\).
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