Continuity, Differentiability and Limits

Continuity at a means that the function value agrees with the common finite limit from both sides. Differentiability means that the limiting tangent gradient exists and is finite. Differentiability implies continuity; continuity alone does not imply differentiability. The graph y=|x| is continuous at zero but has left gradient -1 and right gradient 1, so it is not differentiable there. These ideas are informal; the current guide does not require examination tests of continuity or differentiability. **Behaviour** | **Example** Converges to a finite limit | lim_x to infinity 2x+1x+3=2 Grows without bound | x2 to +infinity as x to infinity No common two-sided limit | 1x to +infinity as x 0+, but 1x to -infinity as x 0-