Continuity and DifferentiabilityIntroduction
Riya rides her scooter to the metro station every morning and leaves it in the station's parking lot. The board at the gate says: ₹20 for every hour or part of an hour. Park for ten minutes and you pay ₹20. Park for one hour and five minutes and you pay ₹40.
Riya wonders what the fee looks like as a graph. Let us draw it with a pen that is only allowed to leave the paper when it absolutely has to.
Drag the clock forward and watch the pen draw the fee. Take it all the way to 4 hours.
Time parked:
Parking fee in ₹ against hours parked. A filled dot is on the graph; an empty ring is not.
The fee graph is a staircase. Between the steps the pen glides along, but at 1, 2 and 3 hours it has to lift off the paper, because the fee jumps by ₹20 in an instant. In those four hours the pen was lifted
Compare that with the distance Riya's metro train has covered since it left the platform. The train cannot be at one place and, an instant later, a kilometre further on, so that graph can be drawn in one stroke. "Drawn without lifting the pen" is the everyday picture of being continuous, and the first job of this chapter is to make it precise.
This chapter carries on from where differentiation stopped in Class XI, where you learnt to differentiate polynomial and trigonometric functions. Here you will meet the ideas of continuity and differentiability and see how they are related. You will learn to differentiate inverse trigonometric functions, and you will meet a new class of functions, the exponential and logarithmic functions, which lead to powerful new techniques of differentiation. Much of this calculus goes back to