Can the derivative of a continuous function always continuous?
Answer: Yes, indeed a derivative does have to be continuous but not at every point.
Can a function have a derivative and not be continuous?
Simply put, differentiable means the derivative exists at every point in its domain. Thus, a differentiable function is also a continuous function. But just because a function is continuous doesn’t mean its derivative (i.e., slope of the line tangent) is defined everywhere in the domain.
Can a derivative function be discontinuous?
The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable. It is possible for a differentiable function to have discontinuous partial derivatives. An example of such a strange function is f(x,y)={(x2+y2)sin(1√x2+y2) if (x,y)≠(0,0)0 if (x,y)=(0,0).
Are all continuous functions derivatives?
No. Since a function has to be both continuous and smooth in order to have a derivative, not all continuous functions are differentiable.
Does the derivative need to be continuous?
Derivative of differentiable function need not be continuous.
What is the meaning of a derivative of a continuous function?
The derivative of a function (if it exists) is just another function. Saying that a function is differentiable just means that the derivative exists, while saying that a function has a continuous derivative means that it is differentiable, and its derivative is a continuous function.
Does derivative need to be continuous?
5.2), the derivative function g2 is thus defined everywhere on R, but g2 has a discontinuity at zero. The conclusion is that derivatives need not, in general, be continuous!
What is not a continuous function?
In other words, a function is continuous if its graph has no holes or breaks in it. For many functions it’s easy to determine where it won’t be continuous. Functions won’t be continuous where we have things like division by zero or logarithms of zero.
What makes a derivative discontinuous?
The derivative of a function at a given point is the slope of the tangent line at that point. A removable discontinuity — that’s a fancy term for a hole — like the holes in functions r and s in the above figure. An infinite discontinuity like at x = 3 on function p in the above figure.
How do you know if a function is continuous?
For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. Discontinuities may be classified as removable, jump, or infinite.
How do you determine if a function is continuous on an interval?
A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(x) satisfies these criteria from x=a to x=b, for example, we say that f(x) is continuous on the interval [a, b].