Question

Prove that if function \( f \) is differentiable at a point \( a \), then it is also continuous at that point.

Hint:

Differentiability at a point implies continuity at that point, but continuity does not necessarily imply differentiability.

Answer 1

Step 1: Understanding differentiability and continuity.
A function \( f \) is said to be differentiable at a point \( a \) if the derivative exists at that point. That is, if the limit \[ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \] exists. If this limit exists, the function is differentiable at \( a \). For the function to be continuous at \( a \), the following condition must hold: \[ \lim_{x \to a} f(x) = f(a). \]

Step 2: Showing that differentiability implies continuity.
Assume that \( f \) is differentiable at \( a \). By definition, we know that: \[ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \text{exists}. \] This is the definition of the derivative of \( f \) at \( a \), which implies that the function has a well-defined rate of change at \( a \). Now, let's prove that \( f \) is continuous at \( a \). For continuity, we need to show that: \[ \lim_{x \to a} f(x) = f(a). \] From the definition of the derivative, we have: \[ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} = L, \] where \( L \) is the derivative of \( f \) at \( a \). This means that as \( h \to 0 \), the difference between \( f(a+h) \) and \( f(a) \) gets closer and closer to zero. Thus, by the definition of continuity: \[ \lim_{x \to a} f(x) = f(a), \] proving that \( f \) is continuous at \( a \).

Step 3: Conclusion.
Therefore, we have proven that if a function \( f \) is differentiable at a point \( a \), then it must also be continuous at that point.