Differentiability and Continuity?

Question

What is the difference between differentiability and continuity of a function? Is it possible that a function not continuous at any particular point is differentiable at that point or vice-versa?

Answer 1

A function can be continuous, but not differentiable. Differentiabilty means: A TANGENT(NOT VERTICAL) can be drawn at a pt.

The graphical condition for CONTINUITY is :
(1) There should not be any break in the graph of function.

The graphical condition for DIFFERENTIABILITY is:
(1) The function MUST be continuous.
(2) The function should not have a CORNER in the graph.
(3) The function should not have a vertical tangent at a point.

Mathematically, these can be expressed as:
Continuity: The limit of the fn should exist at the pt.P and it should be equal to f(P).
Differentiability:
The Right Hand Derivative = The Left Hand Derivative
So, you see; a fn can be continuous but not differentiable; but if a fn is differentiable, it has to be continuous.
Example: f(x)=|x| is continuous at x=0, but not differentiable at x=0. {It has a CORNER}

For exact definition and more information, please go to:
www.mathworld.wolfram.com {If URL is not correct please google it}

Answer 2

Yes. Continuity means that any limit of the function is actually attained on a closed interval. Differentiability requires that the tangent at any point of a continuous cure be unique. An example of a continuous but not differentiable curve is the absolute value function y=|x| on an interval including x=0. It is continuous because the limit as |x|->0 of y is in fact y(0), but there is no unique tangent--the slope at (0,0) i undefined and can formally take any value between -1 and +1.

By the way--sorry for the bad probability answer. It's pretty easy to see how I overcounted.

Answer 3

If a function is differentiable at a point, it is continuous there. But, it is possible for a function to be continuous *everywhere* and differentiable *nowhere*. The first example of such a function was given by Karl Weierstrass. It is known that *most* continuous functions fail to differentiable at any point where 'most' is in the sense of the Baire Category Theorem.

Answer 4

If a function is differentiable at a point, it is continuous at that point. The definition of differentiability guarantees that.

For f(x) to be differentiable at x = a, you need f(a) to be defined and
lim x->a (f(x)-f(a)/(x-a) to exist. Since the denominator of the difference quotient goes to 0, the numerator must also go to 0
That is the same as lim x->a f(x) = f(a), which is what you need for continuity.

f(x) = |x| (absolute value) is continuous on the reals but is not differentiable at x = 0, an example that continuity doesn't require differentiability.

Answer 5

A function has to be continuous to be differentiable, but not all continuous functions are diiferentiable. Hilbert gave an example that intuition was not reliable by showing such a function.

Answer 6

Differentiability is a stronger requirement that continuity. There are examples of functions continuous on an interval which are nowhere differentiable on that interval (see, e.g., "Principles of Mathematical Analysis", by Rudin for an excellent example in Ch. 7) But differentiability on an interval always implies continuity of the function on that interval. (Note that **only** applies for closed intervals ☺)


Doug

Answer 7

i'm now unlikely to flow through all that artwork, yet i am going to talk about that the equation is completely the fee of a function it really is non-stop and differenatiable everywhere. hence, the function is non-stop everywhere and differentiable everywhere except the position f(x)=0. hence, the function is continusous yet not differentiable at x=-2 and is non-stop and differentiable at x=-3.

Answer 8

Any function that is continuous is differentiable. And any differentiable function is continous. A discontinous function will have a section of infinite gradient.

Answer 9

Probably..but you would have consider all the options first, I mean, my suggestion would be to take both possibilities and then discount one in favour of the other and 'voila' there you have it, in a nutshell!

Answer 10

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