Determine if functions are continuous at given points

Published on December 26, 2024

Calculus

The provided examples demonstrate the continuity of functions at specific points. Polynomials are continuous everywhere, functions with undefined points are not continuous at those points, and absolute value functions are continuous everywhere, including at points where the expression inside the absolute value is zero.

Question

please solve the exercise

please show me the solution way with the forms

Answer

Here’s the solution:

1. for

Polynomials are continuous everywhere, so

is continuous at

.

Wahr (True)

2. for

The function is not defined at

because the denominator becomes zero. A function cannot be continuous at a point where it is not defined.

Falsch (False)

3. for

The absolute value function is continuous everywhere. Let’s analyze

near

:

For , .
For , .

The left-hand limit as

approaches 2 is:

.

The right-hand limit as

approaches 2 is:

.

Since both the left and right limits exist and are equal to

, the function is continuous at

.

Wahr (True)

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