Modify the graphs of the functions ƒ and g as shown to obtain functions that are continuous on 3a, b4. yContinuouson [a, b)y5 f(x)Right-continuous at aa(a)Continuouson (a, b]y 5 g(x)Left-continuous at ba(b)

Answered: y Continuous on [a, b) y5 f(x) Right-…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Modify the graphs of the functions ƒ and g as shown to obtain functions that are continuous on 3a, b4.

y
Continuous
on [a, b)
y5 f(x)
Right-
continuous at a
a
(a)
Continuous
on (a, b]
y 5 g(x)
Left-
continuous at b
a
(b)

Expert Solution

Step 1

Given that the graph of functions $y = f (x)$ and $y = g (x)$ are given.

We have to modify the graphs of the functions $f$ and $g$ to obtain functions that are continuous on $[a, b]$ .

Step 2

Continuous of a function:

If a function $f$ is continuous at $x = a$ , then $\lim_{x \to a} f (x) = f (a)$ .

Continuous for the graph $y = f (x)$ :

At the left endpoint,

The graph $y = f (x)$ is defined at $x = a$ and $\lim_{x \to a^{+}} f (x) = f (a)$ .

Thus, the graph $y = f (x)$ is right continuous at $x = a$ .

At the right endpoint,

The graph $y = f (x)$ is undefined at $x = b$ and $\lim_{x \to b^{-}} f (x)$ exits.

To make $y = f (x)$ is left continuous at $x = b$ , we can define $f (b) = \lim_{x \to b^{-}} f (x)$ .

Answer:

To modify the graph at $x = b$ for $f (b) = \lim_{x \to b^{-}} f (x)$ , such that the graph $y = f (x)$ is continuous on the interval $[a, b]$ .

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y Continuous on [a, b) y5 f(x) Right- continuous at a a (a) Continuous on (a, b] y 5 g(x) Left- continuous at b a (b)