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Differentiability implies continuity This is easy to prove using the limit of the difference quotient
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What function is continuous everywhere but not differentiable?
Weistrass function is continuous everywhere but not differentiable everywhere
Is there a function that is continuous everywhere differentiable at rationals but not differentiable at irrationals?
No.
Condition for the continuity and differentiablity of a function?
An intuitive answer (NOTE: this is far from precise!) A function is continuous if you can trace its graph without lifting your pencil from the page. If, additionally, it is smooth everywhere without any jagged edges or abrupt corners, then it is differentiable. It is not possible for a function to be differentiable but not continuous. On the other hand, plenty of functions are continuous without being differentiable.
When finding the derivative of a point on a piecewise function does every function in the piecewise function need to be continuous and approach the same limit?
All differentiable functions need be continuous at least.
Classical optimization methods?
Classical optimization methods are analytical and useful in finding the optimum solution of differentiable and continuous functions. They do have limited scope in practical applications.
When was function not having a derivative at a point?
Definition: A function f is differentiable at a if f'(a) exists. it is differentiable on an open interval (a, b) [or (a, ∞) or (-∞, a) or (-∞, ∞)]if it is differentiable at every number in the interval.Example: Where is the function f(x) = |x| differentiable?Answer:1. f is differentiable for any x > 0 and x < 0.2. f is not differentiable at x = 0.That's mean that the curve y = |x| has not a tangent at (0, 0).Thus, both continiuty and differentiability are desirable properties for a function to have. These properties are related.Theorem: If f is differentiable at a, then f is continuous at a.The converse theorem is false, that is, there are functions that are continuous but not differentiable. (As we saw at the example above. f(x) = |x| is contionuous at 0, but is not differentiable at 0).The three ways for f not to be differentiable at aare:a) if the graph of a function f has a "corner" or a "kink" in it,b) a discontinuity,c) a vertical tangent
What are the seven types of function?
There are infinitely many types of functions. For example: Discrete function, Continuous functions, Differentiable functions, Monotonic functions, Odd functions, Even functions, Invertible functions. Another way of classifying them gives: Logarithmic functions, Inverse functions, Algebraic functions, Trigonometric functions, Exponential functions, Hyperbolic functions.
Where is f(x) discontinuous but not differentiable Explain?
Wherever a function is differentiable, it must also be continuous. The opposite is not true, however. For example, the absolute value function, f(x) =|x|, is not differentiable at x=0 even though it is continuous everywhere.
What are the similarities between quadratic function and linear function?
Both are polynomials. They are continuous and are differentiable.
How is the function differentiable in graph?
If the graph of the function is a continuous line then the function is differentiable. Also if the graph suddenly make a deviation at any point then the function is not differentiable at that point . The slope of a tangent at any point of the graph gives the derivative of the function at that point.
What are some function words?
Domain, codomain, range, surjective, bijective, invertible, monotonic, continuous, differentiable.
What is a fractal and who is famous for discovering the concept?
A fractal is a shape which shows the same pattern at every level of magnification.The concept of fractals emerged form the ideas of recursive functions in the 17 Century. Those ideas were put into more rigorous framework by 19th Century mathematicians like Bolzano, Reimann and Weierstrass who studied functions which were continuous but not differentiable - particularly those that were non-differentiable almost everywhere. The term, "fractal" was first used by Mandelbrot in 1975.
