245A, Notes 5: Differentiation theorems

Differentiation Theorems from Terence Tao

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Let $latex {[a,b]}&fg=000000$ be a compact interval of positive length (thus $latex {-infty < a < b < +infty}&fg=000000$). Recall that a function $latex {F: [a,b] rightarrow {bf R}}&fg=000000$ is said to be differentiable at a point $latex {x in [a,b]}&fg=000000$ if the limit

$latex displaystyle F'(x) := lim_{y rightarrow x; y in [a,b] backslash {x}} frac{F(y)-F(x)}{y-x} (1)&fg=000000$

exists. In that case, we call $latex {F'(x)}&fg=000000$ the strong derivative, classical derivative, or just derivative for short, of $latex {F}&fg=000000$ at $latex {x}&fg=000000$. We say that $latex {F}&fg=000000$ is everywhere differentiable, or differentiable for short, if it is differentiable at all points $latex {x in [a,b]}&fg=000000$, and differentiable almost everywhere if it is differentiable at almost every point $latex {x in [a,b]}&fg=000000$. If $latex {F}&fg=000000$ is differentiable everywhere and its derivative $latex {F’}&fg=000000$ is continuous, then we say that $latex {F}&fg=000000$ is continuously differentiable.

Remark 1Much…

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