Continuity and Differentiability

In summary: The definition I learned in high school, long ago, is that sin x is the y-coordinate of the point on the unit circle with angle x. That is, we define sin x as "the function such that sin x= y where (x,y) is the point on the unit circle with angle x". That is, we first define sin x geometrically and then define the function as the "y" of the point on the unit circle. By that definition, sin x is certainly continuous. The function is also differentiable everywhere with derivative cos x. If you want a more analytic definition, sin x is the "infinite series" x- (1/3!)x^3+ (1/
  • #1
daytrader
4
0

Homework Statement



Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks

1) f(x) = x^(2/3) -1 on [-8,8]

answer: function is continuous but not differentiable on -8.

Is that because once I take the derivative I get -8 under a root?? and why is it continous?

2) f(x) = SinX on [0,2pi]

answer: function continuous and differentiable. Why is it continous?

please provide some example if you can.

thanks
 
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  • #2
Here's how I like to think about it: If I can draw the function with a pencil without lifting it up off the paper, then the function is continuous. However, this does not mean it is necessarily differentiable. For instance, I can draw the absolute value function without lifting my pencil up, but it is not differentiable at x=0.
 
  • #3
If a function is continuous but not differentiable, that means the limit [tex]\lim_{h \to 0}{[f(x+h)-f(x)]/h}[/tex] doesn't exist. Graphically, can have a sharp edge so there are infinitely many lines which can intersect the graph only at that one point (so the left hand limit doesn't equal the right hand limit), or in this case the limit of the slope diverges to infinity (positive infinity from the right and negative infinity from the left). Try graphing it to see what I mean. In addition to functions which are continuous everywhere and not differentiable at a point, there are functions where are continuous everywhere but http://en.wikipedia.org/wiki/Nowhere_differentiable"
 
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  • #4
daytrader said:

Homework Statement



Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks

1) f(x) = x^(2/3) -1 on [-8,8]

answer: function is continuous but not differentiable on -8.

Is that because once I take the derivative I get -8 under a root?? and why is it continous?
There is no problem with taking a third root of -8. I suspect you have copied the answer wrong. f is continuous but not differentiable on the interval [-8,8] because it is not differentiable at x= 0. It certainly is differentiable at x= -8.

2) f(x) = SinX on [0,2pi]

answer: function continuous and differentiable. Why is it continous?

please provide some example if you can.

thanks
What is your definition of sin x?
 

FAQ: Continuity and Differentiability

What is continuity?

Continuity is a property of a function where the output values change smoothly as the input values change. This means that there are no abrupt jumps or breaks in the graph of the function.

What is a continuous function?

A continuous function is a function that is continuous at every point in its domain. This means that the function is defined at all points in its domain and there are no breaks or jumps in its graph.

What is a discontinuous function?

A discontinuous function is a function that is not continuous at one or more points in its domain. This means that there are breaks or jumps in the graph of the function at those points.

What is differentiability?

Differentiability is a property of a function where it has a well-defined derivative at every point in its domain. This means that the function is smooth and has no sharp turns or corners.

What is the difference between continuity and differentiability?

Continuity is a more general concept that only requires the function to be smooth and have no abrupt jumps or breaks in its graph. Differentiability, on the other hand, is a stricter condition that requires the function to have a well-defined derivative at every point in its domain.

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