Let f:R2 to R defined by f(x)=√(x2+y2) show that f is continuous on R2
now, to show that f(x,y) is continuous everywhere on ,
we first fix an arbitrary and .
here note that is indeeded the Euclidean's norm on ,
so we can use the reverse triangle inequality.
For any with
hence, proving is continuous in our arbitrary chosen (a,b).
since, this is our arbitrary chosen ,
we conclude that is continuous everywhere.
this assumes that both domain and co-domain of
are equipped with corresponding euclidean metrics.
hence,proved.
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