consider the function graphed below. at what x-values does the function appear to not be continuous?

Contents:

1. Definition of Differentiable
2. Continuously Differentiable
3. Infinitely Differentiable
4. Non Differentiable Functions
5. Nowhere Differentiable

What is Differentiable?

Differentiable means that a function has a derivative. In simple terms, it means there is a slope (one that you tin can calculate). This slope volition tell you something about the rate of change: how fast or slow an event (like dispatch) is happening.

The derivative must be for all points in the domain, otherwise the function is not differentiable. This might happen when you lot have a hole in the graph: if there's a pigsty, in that location's no gradient (there's a drib off!).

Continuously Differentiable

A continuously differentiable office is a office that has a continuous office for a derivative.

In calculus, the ideal function to work with is the (usually) well-behaved continuously differentiable part. If yous take a function that has breaks in the continuity of the derivative, these tin acquit in strange and unpredictable means, making them challenging or impossible to piece of work with.

Formal Definition

More than formally, a function f: (a, b) → ℝ is continuously differentiable on (a, b) (which tin be written as f ∈ C¹ (a, b)) if the following two conditions are true:

1. The function is differentiable on (a, b),
2. f′: (a, b) → ℝ is continuous.

Where:

• f = a function
• f′ = derivative of a function (′ is prime notation, which denotes a derivative).
• ℝ = the set up of all real numbers ("reals").
• ∈: "Is an chemical element of".
• (a, b): an interval from a to b.

Example

The part f(x) = x³ is a continuously differentiable function because information technology meets the above two requirements.

1. The derivative exists: f′(ten) = 3x
2. The function is continuously differentiable (i.e. the derivative itself is continuous)

Run across likewise: Continuous Derivatives.

Do All Differentiable Functions Have Continuous Derivatives?

When you lot first studying calculus, the focus is on functions that either take derivatives, or don't have derivatives. You may exist misled into thinking that if you can find a derivative so the derivative exists for all points on that function. However, a differentiable function and a continuous derivative do not necessarily go hand in hand: it's possible to have a continuous function with a non-continuous derivative.

One case is the function f(x) = x² sin(1/x). Despite this beingness a continuous function for where we can find the derivative, the oscillations make the derivative role discontinuous. You can find an example, using the Desmos calculator (from Norden 2015) here.

Infinitely Differentiable Role

All polynomial functions are infinitely differentiable (graphed at Desmos.com)

An infinitely differentiable office can be differentiated an uncountable, never ending, number of times. More precisely, if a part f has derivatives f (n): (a, b) → ℝ of all orders due north ∈ N, so f is infinitely differentiable on the open interval (a, b) [1]. "All orders" ways get-go derivative, 2nd deritive, and so on, advertizing infinitum.

Infinitely differentiable function examples: All polynomial functions, exponential functions, cosine and sine functions. Whatever combination, production, or sum of these functions. A specific example is the polynomial function f(x) = xy. Note that at some signal, the derivative will equal zippo, merely that doesn't mean it isn't differentiable: the derivative of 0 is just 0 once more, and so on.

Complex functions are infinitely differentiable if they are differentiable one time; In other words, if yous can find the first derivative of a complex function, then you can observe them all.

On the other hand, an case of a non-infinitely differentiable office is the absolute value function f(x) = |10|; The derivative does not be at 10 = 0.

Infinitely Differentiable Function vs. Polish Part

Sometimes, infinitely differentiable functions are sometimes called polish functions (and vice versa), simply there is a subtle deviation: smoothen functions do non accept to be infinitely differentiable (although they often are). Generally speaking, shine functions have continuous derivatives upwardly to a certain lodge, say the 10th derivative. If a smooth function has derivatives of all orders, up to infinity, and then those polish functions are too infinitely differentiable.

Similarly, an analytic function is an infinitely differentiable function; Infinitely differentiable functions are too ofttimes analytic for all x, but they don't have to exist [ii, 3]. A office divers on a closed interval is analytic, if for every signal x₀, in that location is a respective Taylor series with a positive radius of convergence that converges to f(x) in in the neighborhood of 10₀. A Taylor serial has a derivative of a Taylor series with the same radius of convergence, and it converges uniformly in every closed interval containing x₀; This means that an analytic function is also infinitely differentiable on the specified interval [four].

How to Figure Out When a Role is Not Differentiable

Watch the video for several examples of non differentiable functions:

Non Differentiable Functions

Can't see the video? Click here.

In general, a part is non differentiable for four reasons:

• Corners,
• Cusps,
• Vertical tangents,
• Jump discontinuities.

You lot'll exist able to run across these different types of scenarios past graphing the part on a graphing calculator; the only other way to "see" these events is algebraically. Fifty-fifty if your algebra skills are very stiff, it'southward much easier and faster just to graph the office and look at the behavior.

How to Check for When a Role is Not Differentiable

Step 1: Check to run across if the part has a distinct corner. For example, the graph of f(10) = |10 – one| has a corner at x = 1, and is therefore not differentiable at that point:

Stride ii: Look for a cusp in the graph. A cusp is slightly dissimilar from a corner. You can recall of information technology every bit a type of curved corner. This graph has a cusp at ten = 0 (the origin):

Stride 3: Look for a jump discontinuity. This usually happens in step or piecewise functions. The office may appear to non be continuous. The following graph jumps at the origin.

Pace 4: Check for a vertical tangent. A vertical tangent is a line that runs straight up, parallel to the y-axis.

This graph has a vertical tangent in the middle of the graph at ten = 0.

Limits and Differentiation

Technically speaking, if at that place'due south no limit to the gradient of the secant line (in other words, if the limit does not be at that bespeak), and so the derivative will not exist at that point. The "limit" is basically a number that represents the slope at a point, coming from any direction.

Nowhere Differentiable

A nowhere differentiable office is, maybe unsurprisingly, non differentiable anywhere on its domain. These functions comport pathologically, much like an oscillating discontinuity where they bounce from betoken to bespeak without ever settling downwards enough to calculate a slope at whatsoever point.

Instance of a Nowhere Differentiable Office

Many of these functions exists, just the Weierstrass part is probably the most famous example, as well as being the first that was formulated (in 1872). Named later its creator, Weierstrass, the function (actually a family of functions) came as a total surprise considering prior to its conception, a nowhere differentiable function was idea to be impossible.

Plot of Weierstrass office over the interval [−2, 2]. Like some fractals, the function exhibits self-similarity: every zoom (ruddy circumvolve) is similar to the plot equally a whole.

Many other archetype examples exist, including the blancmange role, van der Waerden–Takagi function (introduced by Teiji Takagi in 1903) and Kiesswetter's function (1966).

The following very unproblematic case of another nowhere differentiable part was constructed by John McCarthy in 1953:

Where: where thousand(x) = 1 + x for −2 ≤ ten ≤ 0, k(ten) = 1 − x for 0 ≤ ten ≤ 2 and g(10) has period iv.

References

[1] Affiliate iv: Differentiable Functions.
[ii] Zhang, F. (2016). Analytic Functions …. Retrieved October 27, 2021 from: https://blogs.ubc.ca/2015math100/2016/03/17/analytic-functions-and-infinitely-differentiable-functions/
[three] Notes on Analytic Functions.
[four] Analytic Functions And Classes Of… Functions: Rice Institute Pamphlet, V29, No. 1, January, 1942
Affiliate iv. Diff. Functions. Retrieved November 2, 2019 from: https://world wide web.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch4.pdf
Karl Kiesswetter, Ein einfaches Beispiel f¨ur eine Funktion, welche ¨uberall stetig und nicht differenzierbar ist, Math.-Phys. Semesterber. 13 (1966), 216–221 (German language)
Larson & Edwards. Calculus.
McCarthy, J. An everywhere continuous nowhere unequal. function. American Mathematical Monthly. Vol. Threescore, No. x, Dec 1953.
Norden, J. Continuous Differentiability. Retrieved November 2, 2015 from: https://www.desmos.com/calculator/jglwllecwh
Desmos Graphing Calculator (images).
Rudin, Due west. (1976). Principles of Mathematical Assay (International Series in Pure and Practical Mathematics) 3rd Edition. McGraw-Hill Instruction.
Su, Francis Eastward., et al. "Continuous just Nowhere Differentiable." Math Fun Facts. .
T. Takagi, A unproblematic example of the continuous function without derivative, Proc. Phys.-Math. Soc. Tokyo Ser. Two ane (1903), 176–177.

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