prove that if the function e^x * sin (1/x) is uniformly continuous on the interval a) (0,1) b) [1/2, 3/2]?

I also know that this function is continuous on (o,infinity) but I don't understand why...

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e^x is uniformly continuous on (-oo,oo).

1/x is uniformly continuous on (-oo,0) U (0,oo).

sin(x) is uniformly continuous on (-oo,oo) this implies sin(1/x) is uniformly continuos on (-oo,0) U (0,oo).

If f(x) and g(x) are uniformly continuous on an interval f(x)*g(x) is uniformly continuous on that same intervarl,

so e^x sin(1/x) is uniformly continuous on (-oo,0) U (0,oo)

so it is certainly uniformly continuous on subintervals of those intervals.

since (0,1) and (1/2, 3/2) are subintervals of (0,oo) e^x sin(1/x) is uniformly continuous on those intervals also.

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