1. Limits and Continuity

Where is the function continuous? Differentiable? Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (0, 3) at which f is not continuous.

Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (-2,2) at which f is not continuous.

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

Continuous Extension

Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2

Limits and Continuity

Repeat the instructions of Exercise 1 for

1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Limits and Continuity

On what intervals are the following functions continuous?

a. ƒ(x) = x¹/³

Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = x⁻²/³

Limits and Continuity

On what intervals are the following functions continuous?

a. ƒ(x) = tan x

Limits and Continuity

On what intervals are the following functions continuous?

b. g(x) = csc x

Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = cos x / x―π

Limits and Continuity

On what intervals are the following functions continuous?

b. g(x) = x³/⁴

Limits and Continuity

On what intervals are the following functions continuous?

d. k(x) = x⁻¹/⁶

Limits and Continuity

On what intervals are the following functions continuous?

d. k(x) = sin x / x

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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