Ucale

# Differentiability in a set

1. A function f(x) defined on an open interval (a,b) is said to be differentiable or derivable in open interval (a,b) if it is differentiable at each point of (a,b)
2. A function f(x) defined on [a,b] is said to be differentiable or derivable at the end points a and b if it is differentiable from the right at a and from the left at b.
In other words $\displaystyle \lim _{ x\longrightarrow { a }^{ + } }{ \frac { f\left( x \right) -f\left( a \right) }{ x-a } } \quad and\quad \lim _{ x\longrightarrow { b }^{ - } }{ \frac { f\left( x \right) -f\left( b \right) }{ x-b } }$

“If f is derivable in the open interval (a,b) and also at the end points a and b, then f is said to be derivable in the closed interval[a,b]”.
“A function f is said to be a differentiable function if it is differentiable at every point of its domain”

April 18, 2019
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