Let $\{f_n\}$ be a sequence of smooth functions which converges to a function $f$. If the convergence is not uniform at a point $a$ the $f$ is discontinuous at $a$. Is there any different type of convergence where if it happens at $a$ then $f$ is continuous at $a$ but is not differentiable.
EDIT
Let $\{f_n\}$ be a sequence of smooth functions which converges to a function $f$.If there is a discontinuity in $f$ at some point $a$ then the convergence is nonuniform.Is there any different type of convergence needed for $f$ to be continuous at some point $a$ but not differentiable ?