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Differentiability of a function

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Differentiability of a function - Lesson Summary

The derivative of a real function f at point c in its domain is defined by:

lim h → 0 f(c+h)-f(c) h

Derivative of f(x) at c = f '(c) or d/dx[f(x)]c

f ' (x) = lim h → 0 f(x+h)-f(x) h  provided the limit exists.

Let y = f(x)

The derivative of y is denoted by y ' or dy/dx.

Differentiation is the process of finding the derivative of a function

If the limit does not exist, it means that function f is not differentiable at c.

A function f is said to be differentiable at point c in its domain, if:

lim h → 0 - f(c + h) - f(c) h    = lim h → 0 + f(c + h) - f(c) h   

A function f is said to be differentiable in interval [a,b], if it is differentiable at every point of [a,b].

f ' (a) is called the right hand derivative and f ' (b) is called the left hand derivative of f.

A function is said to be differentiable in (a,b), if it is differentiable at every point of (a,b).


Theorem: If function f is differentiable at point c, then it is also continuous at that point.

Proof:

Given function f is differentiable at point c.

By definition, f ' (c) = lim x → c f(x) - f(c) x - c

For x ≠ c, we have

f(x) - f(c) = (f(x) - f(c))/(x - c) x (x - c)

lim x → c [f(x) - f(c)]  = lim x → c [ f(x) - f(c) x - c × (x - c)]

⇒ lim x → c f(x) -   lim x → c f(c)   = lim x → c [ f(x) - f(c) x - c ] ×  lim x → c (x - c)

= f ' (c) = 0

⇒ lim x → c f(x) = lim x → c f(c) = f(c)

Hence, f is continuous at x = c.

Note:

All differentiable functions are continuous, but the converse is not always true.

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