We prove that the natural map $H_{\text{b}}^2\left(\Gamma \right)\to H^2\left(\Gamma \right)$ from bounded to usual cohomology is injective if $\Gamma$ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial unitary coefficients, and implies finiteness results for $\Gamma$: the stable commutator length vanishes and any $C^1$–action on the circle is almost trivial. We introduce the continuous bounded cohomology of a locally compact group and prove our statements by relating $H^{*}\text{b}\left(\Gamma \right)$ to the continuous bounded cohomology of the ambient group...