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g(x) = max{ f1(x),...., fN(x)}.

show that g is a continuous function.

I have let t ε [a,b]

NTS g(x) is continous at t

if a≤t< b, then there exists a neighborhood Vt of t contained in [a,t), such that g(x) = f1(x) for any x ε Vt.

Note that f1(x) is continuous on [a,b]. we obtain f1 is continuous on t. → g(x) is is continuous at t.

likewise, if a<t≤b, then there exists a neighborhood Vt of t contained in (t,b], such that g(x) = fN(x) for any x ε Vt.

Note that fN is continuous on [a,b]. fN is continuous on t. g(x) is continuous at t.

by g(x) = f1(x) True, then g(x) = fi+1(x) is also true for all i ε N.

IS this getting close or in the right ballpark? because i also have another more general one.