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

Properties of definite integral, Properties 1.  ∫ b a f ( x ) dx = -∫ ...

Properties 1.  ∫ b a f ( x ) dx = -∫ b a f ( x ) dx .  We can interchange the limits on any definite integral, all that we have to do is tack a minus sign onto the integral

Limits at infinity part ii, Limits At Infinity, Part II :  In this sectio...

Limits At Infinity, Part II :  In this section we desire to take a look at some other kinds of functions that frequently show up in limits at infinity.  The functions we'll be di

Pair of straight line, The equation ax2 + 2hxy + by2 =0 represents a pair o...

The equation ax2 + 2hxy + by2 =0 represents a pair of straight lines passing through the origin and its angle is tan q = ±2root under h2-ab/(a+b) and even the eqn ax2+2hxy+by2+2gx+

Objectives of ones tens and more, Objectives After studying this unit, ...

Objectives After studying this unit, you should be able to 1.  evolve and use alternative activities to clarify the learner's conceptual 2.  understanding of ones/tens/hu

Sketch a graph of the microphone signal, Figure shows noise results for a p...

Figure shows noise results for a prototype van measured on a rolling road. The vehicle had a four-cylinder-in-line engine. The engine speed was varied in 3rd gear from just above

Find the annual percentage yield, 1.   Find the APY for the bank described ...

1.   Find the APY for the bank described below- A bank offers an APR of 4% compounded monthly. 2.  Use the compound interest formula to compute the balance in the following a

Integers, need answer to integers that equal 36

need answer to integers that equal 36

Find out the probability, a)  A husband and wife appear in an interview for...

a)  A husband and wife appear in an interview for two vacancies in the same post.  The probability of husband's selection is 1/7 and that of wife's selection is 1/5.  What is th

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