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### Show that the function f defined by f(x) = xsin(1/x) and 0 when x=0 is continuous at x=0.

Show that the function f defined by f(x) = xsin(1/x) and 0 when x=0 is continuous at x=0.

Show that the function f defined by f(x) = xsin(1/x) and 0 when x=0 is continuous at x=0.

### Which of the following functions? ^{[1]}

Mathematics and Physics; Calculus and Mechanics Specialty. Using this we get ƒ'(x)=(1/3)x(1/3)-1→ƒ'(x)=(1/3)x-2/3→ƒ'(x)=1/(3×2/3)

A vertical asymptote represents dividing by 0, which is not possible, hence the function is not differentiable at (0,0)

### Expert Maths Tutoring in the UK ^{[2]}

A continuous function, as its name suggests, is a function whose graph is continuous without any breaks or jumps. i.e., if we are able to draw the curve (graph) of a function without even lifting the pencil, then we say that the function is continuous

Let us study more about the continuity of a function by knowing the definition of a continuous function along with lot more examples.. A function f(x) is said to be a continuous function in calculus at a point x = a if the curve of the function does NOT break at the point x = a

Is this definition really giving the meaning that the function shouldn’t have a break at x = a? Let’s see. “limₓ → ₐ f(x) exists” means, the function should approach the same value both from the left side and right side of the value x = a and “limₓ → ₐ f(x) = f(a)” means the limit of the function at x = a is same as f(a)

### Definition, Graph and Examples ^{[3]}

In mathematics, a continuous function is a function that does not have discontinuities that means any unexpected changes in value. A function is continuous if we can ensure arbitrarily small changes by restricting enough minor changes in its input

In other words, we can say that a function is continuous at a fixed point if we can draw the graph of the function around that point without lifting the pen from the paper’s plane.. Mathematically, we can define the continuous function using limits as given below:

We can elaborate the above definition as, if the left-hand limit, right-hand limit, and the function’s value at x = c exist and are equal to each other, the function f is continuous at x = c. If the right hand and left-hand limits at x = c coincide, then we can say that the expected value is the limit of the function at x = c

### SOLVED: Which one of the following functions shows that the statement If f is continuous at x 0,then f is differentiable at 0″ is FALSE? ^{[4]}

Get 5 free video unlocks on our app with code GOMOBILE. Which one of the following functions shows that the statement If f is continuous at x 0,then f is differentiable at & 0″ is FALSE?

Which of the following would be a counterexample to the statement: “If $f$ is differentiable at $x=a$ then $f$ is continuous at $x=a ” ?$(a) A function which is not differentiable at $x=a$ but is continuous at $x=a$(b) A function which is not continuous at $x=a$ but is differentiable at $x=a$(c) A function which is both continuous and differentiable at $x=a$(d) A function which is neither continuous nor differentiable at $x=a$. If it is false, give an example to show why it is false.- If $f$ is continuous on $[a, b], f$ is differentiable on $(a, b),$ and $f^{\prime}(x) \neq 0$ for all $x$ in $(a, b),$ then the absolute maximumvalue of $f$ on $[a, b]$ is $f(a)$ or $f(b)$

If $f$ is continuous at $x=a,$ then $f$ is differentiable at $x=a$. The question is that if the state statement is given that even function is continuous at If the function is continuous at X equals to zero, then f is differentiable at zero is a false

### Answered: Which of the following functions does… ^{[5]}

Which of the following functions does NOT satisfy the statement “If a function is continuous at x = 0, then it is differentiable at x = 0”? a. Which of the following functions does NOT satisfy the statement “If a function is continuous at x = 0, then it is differentiable at x = 0”? a

If the graph of a function is symmetric about the ___________, it is called an…. If the values of fx get larger as x increases on an interval, we say that the…

If the values of fx do not change as x increases on an interval, we say that the…. A local ________ occurs where a function changes from decreasing to increasing.

### 2.6: Continuity ^{[6]}

Summary: 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

It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.. Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page

They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs.. We begin our investigation of continuity by exploring what it means for a function to have continuity at a point

### Intermediate value theorem (IVT) review (article) ^{[7]}

– Worked example: using the intermediate value theorem. – Justification with the intermediate value theorem: table

– Justification with the intermediate value theorem. Review the intermediate value theorem and use it to solve problems.

More formally, it means that for any value between and , there’s a value in for which .. This theorem makes a lot of sense when considering the fact that the graphs of continuous functions are drawn without lifting the pencil

### Continuous function ^{[8]}

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities

A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces

### Calculus Volume 1 ^{[9]}

– Explain the three conditions for continuity at a point.. – State the theorem for limits of composite functions.

Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains

We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. Intuitively, a function is continuous at a particular point if there is no break in its graph at that point.

### Continuity and IVT ^{[10]}

The graph shown in Figure 3.3(a) represents a continuous function. Geometrically, this is because there are no jumps in the graphs

For example, we can see that this is not true for function values near \(x=1\) on the graph in Figure 3.3(b) which is not continuous at that location.. Some readers may prefer to think of continuity at a point as a three part definition

A function \(f(x)\) is continuous at \(x=a\) if the following three conditions hold:. \(f(a)\) is defined (that is, \(a\) belongs to the domain of \(f\)),

### Sources

- https://www.wyzant.com/resources/answers/23137/which_of_the_following_functions
- https://www.cuemath.com/calculus/continuous-function/#:~:text=Important%20Notes%20on%20Continuity%3A&text=A%20function%20is%20continuous%20at,All%20polynomials%20are%20continuous.
- https://byjus.com/maths/continuous-function/#:~:text=In%20mathematics%2C%20a%20continuous%20function,minor%20changes%20in%20its%20input.
- https://www.numerade.com/ask/question/which-one-of-the-following-functions-shows-that-the-statement-if-f-is-continuous-at-x-0then-f-is-differentiable-at-0-is-false-98295/
- https://www.bartleby.com/questions-and-answers/which-of-the-following-functions-does-not-satisfy-the-statement-if-a-function-is-continuous-at-x-0-t/bbe06896-dce0-4ab9-8f27-5edd8f1aaddd
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/Chapter_2_Limits/2.6%3A_Continuity
- https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-16/a/intermediate-value-theorem-review
- https://en.wikipedia.org/wiki/Continuous_function
- https://courses.lumenlearning.com/suny-openstax-calculus1/chapter/continuity/
- https://www.sfu.ca/math-coursenotes/Math%20157%20Course%20Notes/sec_ContinuityIVT.html