# Let \((X,d_x)\) and \((Y,d_Y)\) be metric space and let \(f:D(f)\subset X\rightarrow Y\) where \(D(f)\) is the domain of \(f\), then \(f\) is continuous if

Let \((X,d_x)\) and \((Y,d_Y)\) be metric space and let \(f:D(f)\subset X\rightarrow Y\) where \(D(f)\) is the domain of \(f\), then \(f\) is continuous if

—>> given \(\epsilon>0\), there exist \(\delta>0\) such that if \(x\in D(f)\) and \(d_X(x,x_0)<\delta\), then \(d_Y(f(x),f(x_0))<\epsilon.

given \(\epsilon>0\), there exist \(\delta>0\) such that whenever \(d_2(x,a)<\delta\), it follows that \(|f(x)-f(a)|<\epsilon\)

given \(\epsilon>0\), there exist \(\delta>0\) such that whenever \(d_max(x,x_0)>\delta\), it follows that \(d_max_Y(f(x),f(x_0))<epsilon\)

given \(\epsilon>0\), there exist \(\delta>0\) such that \(d_x(x,x_0)<\epsilon\) then \(d_X(f9x),f(x_0))<\delta\)

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