Theorem, from definition of derivative, Mathematics

Assignment Help:

Theorem, from Definition of Derivative

 If f(x) is differentiable at x = a then f(x) is continuous at x =a.

Proof : Since f(x) is differentiable at x = a we know,

f'(a) = lim x→a (f(x) - f(a))/(x - a)

exists. We will require this in some.

 If we next suppose that x ≠ a we can write the as given below,

f(x) - f(a) = ((f(x) - f(a))/( x -a)) (x -a)

Afterward fundamental properties of limits tells us as we have,

lim x→a (f(x) - f(a)) = lim x→a [((f(x) - f(a))/(x - a)) (x -a)]

= lim x→a (f(x) - f(a))/(x - a) lim x→a (x -a)

The primary limit on the right is only f′(a) as we considered above and the second limit is obviously zero and therefore,

lim x→a (f(x) - f(a)) = f'(a).0 = 0

So we've managed to prove as,

lim x→a (f(x) - f(a)) = 0

Although just how does this help us to x= a, prove that f(x) is continuous at x = a?

 Let's establish with the subsequent.

lim x→a (f(x)) = lim x→a [f(x) + f(a) - f(a)]

Remember that we have just added in zero upon the right side. Some rewriting and the utilize of limit properties provides,

limx→a (f(x)) = limx→a [f(a) + f(x) - f(a)]

= limx→a f(a) + limx→a [f(x) - f(a)]

Here, we only proved above that limx→a [f(x) - f(a)] = 0 and since f(a) is a constant we also know that limx→a f(a) = f(a), then it should be,

limx→a f(x) = limx→a f(a) = 0 = f(a)

Or conversely, limx→a f(x) = f(a) although it is exactly what this means for f(x) is continuous at x = a and therefore we are done.


Related Discussions:- Theorem, from definition of derivative

Unite Ratet, How does finding the unit rate help make smart decisions?

How does finding the unit rate help make smart decisions?

Write the value of sin10+sin20+sin30+....+sin360., sin10+sin20+sin30+....+s...

sin10+sin20+sin30+....+sin360=0 sin10+sin20+sin30+sin40+...sin180+sin(360-170)+......+sin(360-40)+sin(360-30)+sin(360-20)+sin360-10)+sin360 sin360-x=-sinx hence all terms cancel

How far did the ?rst arrow goes, From a fixed point directly in front of th...

From a fixed point directly in front of the center of a bull's eye, Kim aims two arrows at the bull's eye. The first arrow nicks one point on the edge of the bull's eye; the second

Complex root - fundamental set of solutions, Example : Back into the comple...

Example : Back into the complex root section we complete the claim that y 1 (t ) = e l t cos(µt)        and      y 2 (t) = e l t sin(µt) Those were a basic set of soluti

Evaluate the area of the shaded region, Evaluate the area of the shaded reg...

Evaluate the area of the shaded region in terms of π. a. 8 - 4π b. 16 - 4π c. 16 - 2π d. 2π- 16 b. The area of the shaded region is same to the area of the squa

Matrices, Consider the following linear equations. x1-3x2+x3+x4-x5=8 -2x1+...

Consider the following linear equations. x1-3x2+x3+x4-x5=8 -2x1+6x2+x3-2x4-4x5=-1 3x1-9x2+8x3+4x4-13x5=49

Integral calculus, I need help to understand: fxx for f(x,y)=x^2+y^2-2xy

I need help to understand: fxx for f(x,y)=x^2+y^2-2xy

Application of probability in business, Application of Probability in Busin...

Application of Probability in Business 1. Business games of chance for illustration, Raffles Lotteries. 2. Insurance firms: this is generally done when a new client or prop

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd