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Question
Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.
Solution
7(x) = x-4; x ≥ 4
= 4 - X ; X<4
`L.H.L : lim_(x->4) f(X)= lim_(x->)4- x =4-4=0`
`R.H.L ; = lim_(x->4) f (X) = lim_(x->4)=4-4=0`
f(4) = 4-4 =0
∴ f(x) is continuous at X = 4.
`R.H.D = f '(4^+) = lim_(h->0) (f(4+h)-f(4))/((4+h)-4)`
= `lim_(h->0) ((4+h-4)-0)/h`
= `lim_(h->0) 1 = 1`
`L.H.D = f'(4^-)=lim_(h->0) (f(4)-f(4-h))/(4- (4 -h))`
`=lim_(h->0) 0 -(4-(4-h))/h `
`=lim_(h->0)- h/h =-1`
∴ L.H.D . ≠R.H.D.
∴ f '(4) does not exists.
∴ f (4) is continuous at X = 4 but non differentiable at X=4.
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