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calculus

Discontinuity

① jump discontinuity

f(x)={2x where x1x+2 where x>1}f(x)= {2x \space where \space x \le 1 \brace -x+2 \space where \space x>1 }

② removable discontinuity

f(x)=(x3)2(x4)x4f(x)= \frac {(x-3)^2(x-4)} {x-4}

③ infinite discontinuity

f(x)=1x2f(x) = \frac {1}{x-2}

④ wild discontinuity

f(x)=cos(1x)f(x) = cos(\frac {1} {x})

Continuity at a Point

a function ff is continuous at x=a{x=a} if

limxafx\lim\limits_{x \to a} fx exists

f(a)f(a) exists

limxafx=f(a)\lim\limits_{x \to a} fx = f(a)

有時就是用第三個表示而已,當作 1, 2 都符合。