Question

You may use a graphing calculator to solve the following problems. True or False If \(f\) has a derivative at \(x=a,\) then \(f\) is continuous at \(x=a .\) Justify your answer.

Short answer

True

Step-by-step solution

Understand differentiability

Firstly, to say that a function \( f(x) \) is differentiable at \( x=a \) means that the derivative \( f'(a) \) exists. This indicates that the function has a defined slope at that point.

Understand continuity

A function \( f(x) \) is said to be continuous at a point \( x=a \) if the limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( f(a) \). It means there are no breaks, holes or jumps at \( x=a \).

Establish the relationship

From Calculus, we know that if a function is differentiable at a point, then the function must also be continuous at that point. This is because differentiability at a point requires the function to be 'smooth' and without corners at that point, which in turn requires the function to be continuous. However, the reverse statement is not necessarily true. A function can be continuous at a point without being differentiable there – for example, at a sharp bend or cusp.

Statethe answer

So, the statement 'If \(f\) has a derivative at \(x=a,\) then \(f\) is continuous at \(x=a\)' is True.