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March 21st, 2018, 06:00 AM   #1
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Range of Continuous & Differentiable Sets

Let $f:R -> R$ and $f(x) = |x-1|+|x-2|$. Let $S1 = ${$x| f$is continuous at $x$} and $S2 = ${$x| f$ is differentiable at $x$} then
a) $S1 = R , S2=R$
b) $S1 = R - ${$1,2$}$, S2 = R$
c) $S1 = R , S2 = R - ${$1,2$}
d) $S1 = R - ${$1,2$}$, S2 = R - ${$1,2$}

Which of the following is true ?

I have seen individual questions like $f(x) = |x-1|$ where it is not continuous at $x=1$. Simillarly, $f(x) = |x-2|$ where it is not continuous at $x=2$.
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March 21st, 2018, 09:01 AM   #2
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for $x \ge 2$ ...

$f(x) = (x-1)+(x-2) = 2x-3$

$f'(x) = 2$

for $1 < x < 2$ ...

$f(x) = (x-1) - (x-2) = 1$

$f'(x) = 0$

for $x \le 1$ ...

$f(x) = -(x-1)-(x-2) = -2x+3$

$f'(x) = -2$


$f(x)$ is continuous over $\mathbb{R}$

$f(x)$ is differentiable over $\mathbb{R} - \{1,2\}$
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March 22nd, 2018, 04:13 AM   #3
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Thank you so much. We have to redefine the function to check continuity & differentiability since the limits have not given. And after checking continuity & differentiability over 1 & 2, it has given the clear answer.
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March 24th, 2018, 06:14 AM   #4
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Quote:
Originally Posted by Lalitha183 View Post
Let $f:R -> R$ and $f(x) = |x-1|+|x-2|$. Let $S1 = ${$x| f$is continuous at $x$} and $S2 = ${$x| f$ is differentiable at $x$} then
a) $S1 = R , S2=R$
b) $S1 = R - ${$1,2$}$, S2 = R$
c) $S1 = R , S2 = R - ${$1,2$}
d) $S1 = R - ${$1,2$}$, S2 = R - ${$1,2$}

Which of the following is true ?

I have seen individual questions like $f(x) = |x-1|$ where it is not continuous at $x=1$. Simillarly, $f(x) = |x-2|$ where it is not continuous at $x=2$.
No, $|x- 1|$ is continuous at x= 1. It is not differentiable there.

If x< 1 then both x- 1 and x- 2 are negative so f(x)= -(x- 1)- (x- 2)= -2x+ 3.

If 1< x< 2 then x- 1 is positive and x- 2 is still negative so f(x)= x- 1- (x- 2)= 1.

If x> 2 then both x- 1 and x- 2 are positive so f(x)= x- 1+ x- 2= 2x- 3.

The limit, as x goes to 1 "from below", is -2(1)+ 3= 1 and "from above" is 1 so f is continuous there. The derivative of f, for x< 1,is just the slope of -2x+ 3, -2. For 1< x< 2, f is constant so the derivative is 0. Those are not the same so f is not differentiable at x= 1.

The limit,as x goes to 2 "from below", is 1 and "from above" is 2(2)- 3= 1 so the function is continuous at x= 2. The derivative of x, for 1< x< 2, is 0. The derivative of f, for x> 2, is 2. Those are not the same so f is not differentiable at x= 2.

(Note: the derivative of a continuous function is NOT necessarily continuous itself but must satisfy the "intermediate value property" so that the two right and left sided slopes not being the same is sufficient to say that the derivative does not exist.)
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