f(x) = |x| cos(1/x), x not equal to 0 also f(x) = 0, x=0 discuss its continuity at x=0.      

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The function `f(x) = |x|*cos (1/x)` for `x!= 0` . Also, f(0) = 0

`lim_(x->0)f(x) = lim_(x->0)(|x|*cos (1/x))`

The value of cos x for any value of x lies in [-1, 1] and the cosine function is an even function with cos(x) = cos(-x)

=> `lim_(x->0)(|x|*cos (1/x)) = 0`

As `f(0) = lim_(x->0)(|x|*cos (1/x)) = 0` , it is continuous at x = 0

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