Question

1: Write an equation that expresses the fact that a function \(f\) iscontinuous at the number \(4\).

Short answer

The required equation is\(\mathop {{\rm{lim}}}\limits_{x \to 4} f\left( x \right) = 4\).

Step-by-step solution

Continuity of a function

A function \(f\) is said to be continuous at a number \(a\) if it satisfies the condition \(\mathop {{\rm{lim}}}\limits_{x \to a} f\left( x \right) = f\left( a \right)\).

Write the equation

It is given that function \(f\) is continuous at the number \(4\). According to the definition of continuous, \(\mathop {{\rm{lim}}}\limits_{x \to 4} f\left( x \right) = f\left( 4 \right)\).

The value of \(\mathop {{\rm{lim}}}\limits_{x \to 4} f\left( x \right)\) exists if and only if \(\mathop {{\rm{lim}}}\limits_{x \to {4^ - }} f\left( x \right) = \mathop {{\rm{lim}}}\limits_{x \to {4^ + }} f\left( x \right)\). This implies that,

\(\begin{array}{c}\mathop {{\rm{lim}}}\limits_{x \to {4^ - }} f\left( x \right) = \mathop {{\rm{lim}}}\limits_{x \to {4^ + }} f\left( x \right)\\ = f\left( 4 \right)\end{array}\)

Thus, the required equation is \(\mathop {{\rm{lim}}}\limits_{x \to 4} f\left( x \right) = 4\).