Continuity at a Point - Calculus

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Questions and Answers

What condition must be satisfied for a function $f$ to be continuous at a point $c$?

$ ext{CHL} eq ext{RHL}$
$ ext{RHL} = f(c)$
$ ext{CHL} = 0$
$ ext{CHL} = ext{RHL} = f(c)$ (correct)

Which of the following correctly describes the right-hand limit of a function at a point $c$?

$ ext{RHL} = ext{CHL}$
$ ext{RHL} = rac{f(c)}{c}$
$ ext{RHL} = ext{lim}_{x o c^{+}}f(x)$ (correct)
$ ext{RHL} = f(c - ext{small})$

Which of the following statements about continuity is incorrect?

For continuity, both limits must equal $f(c)$.
If $ ext{CHL} eq ext{RHL}$, $f$ cannot be continuous at $c$.
A function can be continuous even if $ ext{CHL} = f(c)$.
Continuity can occur at a point where $f(c)$ is undefined. (correct)

What does the term 'left-hand limit' refer to in relation to a function at point $c$?

<p>$ ext{lim}_{x o c^{-}}f(x)$ (D)</p> Signup and view all the answers

What can be concluded if $ ext{CHL} = f(c)$ and $ ext{RHL} eq f(c)$?

<p>The function is not continuous at $c$. (A)</p> Signup and view all the answers

Flashcards

Continuity at a Point

A function is continuous at a point if the limit of the function as x approaches the point is equal to the function's value at the point.

Left-hand Limit

The limit of a function as x approaches a point from the left-hand side.

Right-hand Limit

The limit of a function as x approaches a point from the right-hand side.

Continuity Requirements

A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point are all equal.

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CHL = RHL = f(c)

For a function to be continuous at a point, the left-hand limit must equal the right-hand limit, which in turn must equal the function's value at that point.

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Study Notes

Continuity at a Point

A function, f, is continuous at a point c if the limit of f(x) as x approaches c is equal to f(c).
This can be broken down into two parts: the left-hand limit and the right-hand limit.
Left-hand limit (LHL) : lim (x→c⁻) f(x)
Right-hand limit (RHL): lim (x→c⁺) f(x)
For a function to be continuous at a point c, the limit from the left and the limit from the right must exist and be equal to the function's value at c, and f(c) must be defined.
Mathematically, this is written as:
- LHL = RHL = f(c)