Question

what is differentiability sugano

Answer 1

Explanation:

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. ... This means that the graph of f has a non-vertical tangent line at the point (x0, f(x0))

More generally, if x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f may also be called locally linear at x0, as it can be well approximated by a linear function near this point.

A function {\displaystyle f:U\subset \mathbb {R} \to \mathbb {R} } {\displaystyle f:U\subset \mathbb {R} \to \mathbb {R} }, defined on an open set {\displaystyle U} U, is said to be differentiable at {\displaystyle a\in U} {\displaystyle a\in U} if any of the following equivalent conditions is satisfied:

The derivative {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}} {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}} exists.

There exists a real number {\displaystyle L} L such that {\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)-Lh}{h}}=0} {\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)-Lh}{h}}=0}. The number {\displaystyle L} L, when it exists, is equal to {\displaystyle f'(a)} f'(a).

There exists a function {\displaystyle g:U\subset \mathbb {R} \to \mathbb {R} } {\displaystyle g:U\subset \mathbb {R} \to \mathbb {R} } such that {\displaystyle f(a+h)=f(a)+f'(a)h+g(h)} {\displaystyle f(a+h)=f(a)+f'(a)h+g(h)} and {\displaystyle \lim _{h\to 0}{\frac {g(h)}{h}}=0} {\displaystyle \lim _{h\to 0}{\frac {g(h)}{h}}=0}.

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