Question

Write an equation that expresses the fact that a function f is continuous at the number 4.

Answer 1

A function f is continuous at number 4 if f(4) is defined, the limit of f(x) as x approaches 4 exists, and this limit is equal to f(4). The equation expressing continuity at x=4 is ℝ(f(4)) = f(4), where ℝ denotes the limit as x approaches 4.

Step-by-step explanation:

To express that a function f is continuous at the number 4, we can write an equation based on the definition of continuity at a point. A function is continuous at a point x = a if three conditions are met:

The function f(a) is defined,

The limit of f(x) as x approaches a is equal to f(a).

Therefore, the equation that represents the continuity of the function f at x = 4 can be written as:

ℝ(f(4)) = f(4)

Where ℝ denotes the limit as x approaches 4.

Answer 2

Use the definition of continuity.
A function is continuous at a point x = a iff
lim [x → a] f(x) = f(a)

In other words, a function is continuous at a point if its value is equal to its limit at that point.

So, for f(x) to continuous at x = 4 we must have
lim f(x) = f(4)
x → 4