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Chapter 11: Q. 6 (page 859)

Let y = f (x). State the definition for the continuity of the function f at a point c in the domain of f .

Short Answer

Expert verified

$The continuity or discontinuity of f at a point' c' can be said only when' c' is in the domain of f$

Step by step solution

01

Step 1. Given information is:

A function y = f(x)

02

Step 2. Checking continuity of function f

$Consider y = f (x) The objective is to define the continuity of the function f at a point c in the domain of f . Definition : R is the set of real numbers . Let E \subseteq R, f : E \to R and c \in E, then we say that f is continuous at c is given ε > 0 there exists a δ > 0 such that |f (x) - f (c)| < ε whenever x \in E and |x - c| < δ . If f is not continuous at c, then we say that f is discontinuous at c . Note : the continuity or discontinuity of f at a point' c' can be said only when' c' is in the domain of f . For example : A polynomial function defined by f : R \to R and f (x) = - 5 x + 10 is continuous at every point of R .$

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Most popular questions from this chapter

For constants $α, β$ , and $γ$ , the graph of a vector-valued function of the form

$r (t) = α e^{γ t} (\cos t) i + α e^{γ t} (\sin t) j + β e^{γ t} k, t \geq 0$

Given a differentiable vector-valued function r(t), what is the definition of the unit tangent vector T(t)?

$Complete the following definition : If r (t) = <x (t), y (t), z (t)> is a differentiable position function, then the velocity vector v (t) is . . . .$

Find and graph the vector function determined $r (t) = ⟨ x (t), y (t) ⟩$ by the differential equation

role="math" localid="1649566464308" $x^{'} (t) = 1 + x^{2}, y^{'} (t) = x^{2}, x (0) = 0, y (0) = 1$ . ( HINT: Start by solving the initial-value problemrole="math" localid="1649566360577" $x^{'} (t) = 1 + x^{2}, x (0) = 0$ .)

$Let r (t) = <x (t), y (t), z (t)> . What is the definition of \lim_{t \to c} r (t) ?$

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