Continuity vs Differentiability

Calculus 1 · Axiom Academy

LESSON Continuity vs Differentiability Every differentiable function is continuous — but a continuous function can still fail to be differentiable. 1. Differentiable Forces Continuous Start at a point where the curve is smooth — it has one clear tangent line. As the point P slides along, watch the tangent tilt to match the slope f'(x) . Crucially, to even have a tangent, the curve must arrive at P without a gap or a jump. So the moment a derivative exists, continuity comes free. A derivative exists at x=a (one tangent slope) ...which forces continuity at x=a 2. Continuous Is Not Enough: the Corner |x| Does the arrow run both ways? No — and f(x)=|x| is the witness. You can trace it through x=0 without lifting your pen, so it is continuous there. But watch a tangent line try to glue itself to the corner: coming from the left the slope is -1 , from the right it is +1 . The two never agree, so there is no single tangent at x=0 — the function is continuous but not differentiable . For x < 0 the graph is the line y=-x , so the slope is -1 . For x > 0 the graph is the line y=x , so the slope is +1 . Since , the difference quotient has no limit at 0 . 3. Break the Curve, Lose the Derivative

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