Question

True or False If the left-hand derivative and the right-hand derivative of \(f\) exist at \(x=a,\) then \(f^{\prime}(a)\) exists. Justify your answer.

Short answer

True. If the left-hand derivative and the right-hand derivative of \(f\) exist and are equal at \(x=a\), then the derivative \(f'(a)\) exists.

Step-by-step solution

Understanding Differentiability

The derivative of a function at a certain point provides the slope of the tangent line to the function at that point. A function \(f\) at a point \(a\) is differentiable if the limit of the difference quotient exists. This requires that the left-hand limit and the right-hand limit are equal.

Left-hand and Right-hand Derivatives

The left-hand derivative and the right-hand derivative are the rates of change of the function as \(x\) approaches point \(a\) from the left and from the right respectively. Essentially, they represent the slopes of the tangent lines to the function at \(x=a\) from the left and from the right. Formally, the left-hand derivative, denoted as \(f'_{-}(a)\), is given by \(\lim_{h->0^-} \frac{f(a+h)-f(a)}{h}\) while the right-hand derivative, denoted as \(f'_{+}(a)\), is given by \(\lim_{h->0^+} \frac{f(a+h)-f(a)}{h}\)

Condition for Existence of \(f'(a)\)

The derivative of the function \(f\) at the point \(a\), denoted by \(f'(a)\), exists only if the left-hand derivative and the right-hand derivative at \(x=a\) are equal. Formally, \(f'(a)\) exists if \(f'_{-}(a) = f'_{+}(a)\).