If f is differentiable at a, must f be continuous at a?

검증된 단계별 안내

Step 1: Understand the definitions. A function f is differentiable at a point a if the derivative f'(a) exists. This means that the limit of the difference quotient exists as x approaches a.

Step 2: Recall the definition of continuity. A function f is continuous at a point a if the limit of f(x) as x approaches a is equal to f(a).

Step 3: Connect differentiability and continuity. If f is differentiable at a, it implies that the limit of the difference quotient exists, which requires that the limit of f(x) as x approaches a must equal f(a).

Step 4: Conclude the relationship. Since differentiability at a point a implies that the function must be continuous at that point, f must be continuous at a if it is differentiable at a.

Step 5: Summarize the result. Differentiability at a point a implies continuity at that point, but the converse is not necessarily true; a function can be continuous at a point without being differentiable there.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Differentiability

A function f is said to be differentiable at a point a if the derivative f'(a) exists. This means that the limit of the difference quotient as x approaches a must exist. Differentiability implies that the function has a well-defined tangent at that point, which is a stronger condition than continuity.

Continuity

A function f is continuous at a point a if the limit of f(x) as x approaches a equals f(a). This means there are no jumps, breaks, or holes in the graph of the function at that point. Continuity ensures that small changes in x result in small changes in f(x), allowing for a smooth transition.

Relationship Between Differentiability and Continuity

If a function is differentiable at a point a, it must also be continuous at that point. This is because the existence of the derivative requires that the function does not have any discontinuities at a. However, the converse is not true; a function can be continuous at a point without being differentiable there.