Continuity & Differentiability Questions with Solutions

A function is continuous at a point if you can draw its graph through that point without lifting your pen. Formally, f(x) is continuous at x = a if the left-hand limit, the right-hand limit and the value f(a) all exist and are equal. A function is differentiable at a point if its derivative exists there, which means the left-hand and right-hand derivatives are equal. This topic covers checking continuity and differentiability at a point and over an interval, the types of discontinuity (removable and jump), and the algebra of continuous and differentiable functions. The most important link is that every differentiable function is continuous, but a continuous function need not be differentiable. Graphs with a sharp corner, such as |x| at x = 0, are continuous but not differentiable there. The most common exam question gives a piecewise function with an unknown constant k and asks for the value that makes it continuous or differentiable at a point. Find the limit or derivative from each side, set them equal, and solve for k. Polynomial, exponential, sine and cosine functions are continuous everywhere, so you only need to check the points where a piecewise definition changes or where a denominator becomes zero. Practise the continuity and differentiability questions below, each with a step-by-step solution.