Question

Show that the function f defined by f(x) = `{{:(x sin  1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.

Answer 1

Left hand limit at x = 0 is given by

`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) x sin  1/x` = 0  ....`["since", -1 < sin  1/x  < 1]`

Similarly `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) x sin  1/x` = 0.

. Moreover f(0) = 0.

Thus `lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) "f"(x)`

= f(0)

Hence f is continuous at x = 0.