Continuity and differentiability are two very important attributes associated with functions. Certain functions are continuous from a certain point to a certain point, but not differentiable throughout. Fine. But what do we mean by continuity and differentiability?
Let us first talk about the graph of a function.
Graph of a function? What is that?
Graph of a function is nothing but a drawing to represent it. We can always draw the graph of a function. It can be in the shape of a curve that can be drawn by hand on a piece of paper or in the form of a straight line or broken lines or a combination of curves and straight line segments or even a scattering of isolated points.
If we can draw the graph of a function from point A to point B without lifting the pencil off the paper, we say that the function is continuous from A to B. That means, the drawing that represents a continuous function has no gaps in it. If there is a gap then the function is not continuous!
Well, that is fair enough. Now, what is the meaning of differentiability?
If the drawing (graph) of a function is continuous from point A to point B and has no sharp bend or sharp turning on it, we say that the function is differentiable throughout between A and B.
But if the graph has gaps or sharp bends or sharp turnings, the function is said to be not differentiable there. In other words a function is not differentiable at the gaps or sharp bends or sharp turnings of the graph.
To summarize, a function is said to be
- continuous throughout if there is no gap in the drawing (graph) of the function
- differentiable throughout if there is no gap, no sharp bends or no sharp turnings in the drawing.
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