The proof of function continuity Given that f (x) and G (x) are continuous at x0, it is proved that h (x) = max (f (x), G (x)) is continuous at x0

The proof of function continuity Given that f (x) and G (x) are continuous at x0, it is proved that h (x) = max (f (x), G (x)) is continuous at x0

First of all, if f (x) is continuous at a certain point, it is easy to prove that | f (x) | is also continuous at that point
And H (x) = (f (x) + G (x) + | f (x) - G (x) | / 2
So h (x) is continuous at x0

A proof of function continuity
If f (x) is continuous at x = 0, and f (x + y) = f (x) + F (y), it holds for any x, y ∈ (- infinite, + infinite), try to prove that f (x) is a continuous function on (- infinite, + infinite)

When x = 0, f (y) = f (0) + F (y)
Then f (0) = 0
Since f (x) is continuous at x = 0, then f (x) - > 0 (X -- > 0)
For any
F (x + Δ x) - f (x) = f (Δ x) - > 0 when Δ x-- > 0
So the continuity of F (x) is proved

How to prove the continuity of function on closed interval

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