Consider f:R→R , f is differentiable at xo , but f' is not continuous at xo
fix xo, MVT tells us
f(x)-f(xo)
───── = f'(c(x)) , where xo─c(x)─x (i.e. c由x而變動)
x - xo
Now we let left hand side A(x) , right hand side B(x)
we know A(x)=B(x) , for all x€R\{xo}
But
lim A(x) = f'(xo)
x→xo
如果藉此我們說因為左邊極限存在
右邊也要跟著如此
所以
lim B(x) 應該要 = f'(xo)
x→xo
可是
lim B(x) = lim f'(c(x)) = f'(xo)
x→xo x→xo
這個等號只成立在f' is continuous at xo
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