Discuss the continuity and differentiability of the function f(x) = 
,
(where m, n ∈ N) at rational and irrational points.
Text Solution
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discontinuous and non-differentiable
Sol. 
is 1 or 0 according to x is a rational number or an irrational number. As m! π x will become integral multiple of π when x is rational, then cos (m! π x) = ± 1. And when m! π x is not an integral multiple of π i.e. when x is irrational then –1 < cos (m! π x) < 1
∴ f(x) = 
∴ f(x) is discontinuous and non-differentiable at every real number.
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