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[SOLVED] Real analysis - show convex functions are left & right differentiable
Let f:R-->R be convex. Show f admits in every point a left derivative and a right derivative.
A function f:R-->R is convex if x1 < x < x2 implies
[tex]f(x)\leq \frac{x_2-x}{x_2-x_1}f(x_1)+\frac{x-x_1}{x_2-x_1}f(x_2)[/tex]
Or equivalently, if whatever x, y, and [itex]\lambda[/itex] in [0,1],
[tex]f(\lambda x + (1-\lambda)y\leq \lambda f(x) + (1-\lambda)f(y)[/tex]By left derivative at x0, we mean the limit
[tex]D_lf(x_0)\lim_{x\rightarrow x_0^-}\frac{f(x)-f(x_0)}{x-x_0}[/tex]
and by right derivative at x0, we mean the limit
[tex]D_rf(x_0)\lim_{x\rightarrow x_0^+}\frac{f(x)-f(x_0)}{x-x_0}[/tex]
Let's stick to the left derivative.
I know convex functions are Lip****z, so the differential quotient is bounded.
I have also proven in an earlier exercise that the differential quotient is increasing as x increases:
"If x1 < x < x2, then [tex]\frac{f(x)-f(x_1)}{x-x_1}\leq \frac{f(x_2)-f(x_1)}{x_2-x_1} \leq\frac{f(x_2)-f(x)}{x_2-x}[/tex]"
But this does not give the conclusion because I must show the differential quotient converges for any sequence, monotonous or not, converging to x0.
Can we show limsup=liminf? Can we show it is Cauchy? I don't see how.
Homework Statement
Let f:R-->R be convex. Show f admits in every point a left derivative and a right derivative.
Homework Equations
A function f:R-->R is convex if x1 < x < x2 implies
[tex]f(x)\leq \frac{x_2-x}{x_2-x_1}f(x_1)+\frac{x-x_1}{x_2-x_1}f(x_2)[/tex]
Or equivalently, if whatever x, y, and [itex]\lambda[/itex] in [0,1],
[tex]f(\lambda x + (1-\lambda)y\leq \lambda f(x) + (1-\lambda)f(y)[/tex]By left derivative at x0, we mean the limit
[tex]D_lf(x_0)\lim_{x\rightarrow x_0^-}\frac{f(x)-f(x_0)}{x-x_0}[/tex]
and by right derivative at x0, we mean the limit
[tex]D_rf(x_0)\lim_{x\rightarrow x_0^+}\frac{f(x)-f(x_0)}{x-x_0}[/tex]
The Attempt at a Solution
Let's stick to the left derivative.
I know convex functions are Lip****z, so the differential quotient is bounded.
I have also proven in an earlier exercise that the differential quotient is increasing as x increases:
"If x1 < x < x2, then [tex]\frac{f(x)-f(x_1)}{x-x_1}\leq \frac{f(x_2)-f(x_1)}{x_2-x_1} \leq\frac{f(x_2)-f(x)}{x_2-x}[/tex]"
But this does not give the conclusion because I must show the differential quotient converges for any sequence, monotonous or not, converging to x0.
Can we show limsup=liminf? Can we show it is Cauchy? I don't see how.
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