4-3. For

f:(a,b)->R

and

xin(a,b)

we define the left-sided derivative of

f

at

x

via the left limit

D^(l)f(x):=\\\\lim_(z->x^(-))(f(z)-f(x))/(z-x)

if the limit exists and we define the right-sided derivative of

f

at

x

via the right limit

D^(r)f(x):=\\\\lim_(z->x^(+))(f(z)-f(x))/(z-x)

if the limit exists.\ (a) Prove that

f

is differentiable at

x

iff the left-sided and right-sided derivatives exist at

x

and

D^(r)f(x)=D^(l)f(x)

.\ (Compare with Standard Proof Technique 3.20.)\ (b) Prove that for

f(x)=|x|

we have that

D^(r)f(0)=1

and

D^(l)f(0)=-1

.

3. For f:(a,b)R and x(a,b) we define the left-sided derivative of f at x via the left limit Dlf(x):=limzxzxf(z)f(x) if the limit exists and we define the right-sided derivative of f at x via the right limit Drf(x):=limzx+zxf(z)f(x) if the limit exists. (a) Prove that f is differentiable at x iff the left-sided and right-sided derivatives exist at x and Drf(x)=Dlf(x). (Compare with Standard Proof Technique 3.20.) (b) Prove that for f(x)=x we have that Drf(0)=1 and Dlf(0)=1