Is Pointwise Convergence Continuous?

Thus, pointwise convergence does not, in general, preserve boundedness. f(x) = { 0 if 0 ≤ x < 1, 1 if x = 1. Although each fn is continuous on [0, 1], their pointwise limit f is not (it is discon- tinuous at 1). Thus, pointwise convergence does not, in general, preserve continuity.

Are pointwise functions continuous?

A piecewise function is continuous on a given interval in its domain if the following conditions are met: ... there is no discontinuity at each endpoint of the subdomains within that interval.

What pointwise continuous?

A function which is continuous at all points in X, but not uniformly continuous, is often called pointwise continuous when we want to emphasize the distinction. Example 1 The function f : R → R defined by f(x) = x2 is pointwise continuous, but not uniformly continuous.