Let
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I’d like to compute explicitly the linearly determined function such that whenever . I want to show its well known continuity.
Let , and . Let
I’d like to find such that , i.e. . This last equation is equivalent to the following ones:
Define . Both summands are non-negative (since ). Thus is a non-negative real number. The two solutions of the above quadratic equation are given by
Thus (of course), and ⇔ — indeed, when then
and it’s clear that . The inverse implication is simple.