陈述：如果函数 f ( x ) {\displaystyle f(x)} 在点 x 0 , {\displaystyle x_{0},} 可导，则 f {\displaystyle f} 在 x 0 . {\displaystyle x_{0}.} 连续。
证明：假设 f ( x ) {\displaystyle f(x)} 在 x 0 {\displaystyle x_{0}} 可导。则 D x f ( x 0 ) {\displaystyle D_{x}f(x_{0})} 存在。
函数 f ( x ) {\displaystyle f(x)} 在 x 0 {\displaystyle x_{0}} 连续，当 lim x → x 0 f ( x ) = f ( x 0 ) . {\displaystyle \lim _{x\to x_{0}}f(x)=f(x_{0}).} ⇒ lim x → x 0 f ( x ) − f ( x 0 ) = 0 {\displaystyle \Rightarrow \lim _{x\to x_{0}}f(x)-f(x_{0})=0} ⇒ lim x → x 0 f ( x ) − lim x → x 0 f ( x 0 ) = 0 {\displaystyle \Rightarrow \lim _{x\to x_{0}}f(x)-\lim _{x\to x_{0}}f(x_{0})=0} ⇒ lim x → x 0 [ f ( x ) − f ( x 0 ) ] = 0. {\displaystyle \Rightarrow \lim _{x\to x_{0}}[f(x)-f(x_{0})]=0.} 因此，为了证明该定理，只需证明 lim x → x 0 [ f ( x ) − f ( x 0 ) ] = 0. {\displaystyle \lim _{x\to x_{0}}[f(x)-f(x_{0})]=0.} 。
lim x → x 0 [ f ( x ) − f ( x 0 ) ] {\displaystyle \lim _{x\to x_{0}}[f(x)-f(x_{0})]} = lim x → x 0 [ f ( x ) − f ( x 0 ) x − x 0 ( x − x 0 ) ] {\displaystyle =\lim _{x\to x_{0}}[{\frac {f(x)-f(x_{0})}{x-x_{0}}}(x-x_{0})]} = lim x → x 0 [ f ( x ) − f ( x 0 ) x − x 0 ] ⋅ lim x → x 0 ( x − x 0 ) {\displaystyle =\lim _{x\to x_{0}}[{\frac {f(x)-f(x_{0})}{x-x_{0}}}]\cdot \lim _{x\to x_{0}}(x-x_{0})} = D x f ( x 0 ) ⋅ 0 = 0 {\displaystyle =D_{x}f(x_{0})\cdot 0=0}
因此，当 D x f ( x 0 ) {\displaystyle D_{x}f(x_{0})} 存在时， lim x → x 0 [ f ( x ) − f ( x 0 ) ] = 0 {\displaystyle \lim _{x\to x_{0}}[f(x)-f(x_{0})]=0} 成立。
∴ {\displaystyle \therefore } 如果 f ( x ) {\displaystyle f(x)} 在点 x 0 , {\displaystyle x_{0},} 可导，则 f {\displaystyle f} 在 x 0 . {\displaystyle x_{0}.} 连续。[证明完毕]
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![{\displaystyle \lim _{x\to x_{0}}[f(x)-f(x_{0})]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e78a34d9aab5d734c394715ac8792864609bcf5f)
